Amanda Levete made her reputation working with in the influential architectural practice, Future Systems. She talks about her 'spectacular failures', and also her many thrilling triumphs

Amanda Levete is showing me a model of her most spectacular failure. We're standing in our stocking feet (her office, her rules) before a little box containing her and Anish Kapoor's 2002 design for the Princess Diana memorial fountain. It consists of a dinky red pillow lying in a model of the Serpentine in London's Hyde Park. White marble steps on one bank sweep down the water's edge to provide a viewpoint.

"It was so beautiful – a blood red pillow that would shoot a 15ft high dome of red water. We wanted to create a wonderful, ethereal place."

Pillows? Blood? Some critics were livid. How dare Levete's architectural practice Future Systems and Kapoor be so insensitive to the memory of her Di-ness as to produce a design that reminded them (poor flowers) of sex and death?

"The judges hated it," recalls Levete. "They asked 'Why red? Why not green?' Anish replied grandly: 'As an artist, I could never work in green.'"

"I was really pissed off we didn't get it," says Levete. But surely, I suggest, she's well out of it. Look what happened to Kathryn Gustafson's winning design: her ring of bright water faced a tsunami of press criticism; visitors injured themselves on the slippery granite or washed their dogs in streams designed for moody contemplation.

Better, sometimes, for architecture to remain unbuilt than be sullied by realisation. This isn't a trite point. It goes to the heart of Levete's formative architectural experiences. At the Architectural Association in the 1970s, Levete was taught by architects who preferred their projects to be hypothetical. "Not one of them, people like Rem Koolhaas and Nigel Coates, intended to build. When I left, I didn't know anything about building." Isn't that nuts? "You could argue it's a problem, but it's also not: it's the one moment you get to explore your creativity. I learned how to build later."

It also goes to the heart of her working relationship with her late ex-husband, Czech architect Jan Kaplický, with whom she designed two of the most remarkable pieces of recent British architecture: the 1998 Media Centre at Lord's cricket ground in London and the 2003 Selfridges department store in Birmingham. "Jan would have been happy not to build. He knew his place in history was assured through his drawings. He couldn't bear to visit the actual buildings. At Selfridges' opening, he stormed off because the finished structure wasn't as pure as the original work."

Levete, though, is more pragmatic. "I don't devalue the power of conceptual thinking, but for me the thrill of architecture is to see your ideas realised. To struggle against the problems out there and overcome them."

For Levete, 55, that creative struggle with an external constraint is one of the things that seduced her into studying architecture in the first place. "After I got expelled from school for sunbathing naked on the roof during a biology lesson at 16, I didn't know what to do. I got so embarrassed that all my friends were going to university that I did an A-level in art and art history, and a foundation year at art school. That's when architecture came across my radar, and when it did, I realised that I work best when I'm doing something creatively, but have a boundary to push. As an artist you have to create your own boundaries. I realised I would find that difficult, whereas architecture is creative, but it has the reality of boundaries you don't create."

But sometimes those boundaries have proven insurmountable. Another of what she calls her "spectacular failures" was a recent project for the Louvre in Paris. Her design envisaged freeing the subterranean space beneath IM Pei's transparent pyramid from its role as holding pen for angry, queueing tourists. "We wanted to create space where visitors could have a moment of repose and think about what they've seen, rather than a clogged entrance hall."

But, again, her ideas were not well received. "The judges told me, 'You're not playing the game.' I knew enough French to say: 'I didn't realise it was a game.' So bureaucratic! For me, architecture is about not playing a game by the rules, it's about challenging the brief you're given – pushing boundaries."

Enough about Levete's (alleged) failures. We're meeting because Amanda Levete Architects has just won the competition to built an extension to the Victoria and Albert Museum. It will be the biggest new art space in London since Tate Modern – a 1,500 sq metre gallery for temporary exhibitions with a new entrance to the building.

Isn't it a poisoned chalice? Seven years ago the V&A abandoned Daniel Libeskind's provocative Spiral extension plan. "It had got through planning and then there was a storm that made the V&A change its mind," says Levete. "But, no, I don't think that will happen to us." That storm included journalist William Rees-Mogg describing Libeskind's plan as a "disaster for civilisation". What does Levete think of Libeskind's plan? "It was iconic, but the time for iconic buildings has passed." Levete met the V&A's new brief by producing a subtler, indeed scarcely perceptible, piece of architecture than Libeskind's strutting, jutting extension, one she argues will create an "iconic space rather than be an iconic building".

Her design takes its cue from the local authority's proposed pedestrianisation of Exhibition Road. "That street will be thronged with people. Our idea is to encourage them to drift in. We want to break down the separation between street and museum. We will draw visitors in from Exhibition Road through a colonnade into a large, light-filled public courtyard, and down into the galleries."

Levete says a lot of the thinking that went into the failed Louvre bid was recycled for the V&A project. There, too, she was concerned with flows of people and light into a subterranean space. "The gallery space can either be flooded with dramatic daylight, or the glass painted black to provide the low light levels that the V&A needs for the delicate materials they sometimes exhibit."

Her aim, she says, in architecture, is to change the way its users interact. "The point of architecture is to contribute to the culture of a city or the culture of a nation. Architecture changes the way you see yourself, the way others see you. It should be respected for that."

But it often isn't. Levete is furious about education secretary Michael Gove's disparaging remarks about her profession. He recently told a conference, "we won't be getting any award-winning architects" to design new schools, "because no one in this room is here to make architects richer".

"I do find it depressing he thinks we're in it to get our snouts in the trough." But does it matter if our kids are educated in schools that look like out-of-town Tescos, so long as they can add up and speak proper? "There's no necessary relationship between how beautiful school buildings are and exam results, but what Gove is saying is: let's have more mediocrity, more crap buildings, because they don't matter, right?

"Already 80% of the profession are not good. You only have to look around London to see that. Politicians too rarely root out the crap. When I think of all the mediocrity in an area of expensive real estate like the City of London, and think how little a genius like Jan – and I don't use the term lightly – saw built in his lifetime, you can't help but think two things: one, the dice are loaded against great architecture; two: work harder."

She met Kaplický in the late 1980s. He was tall, elegant, handsome and "very Czech, by which I mean passionate and pessimistic" – the very embodiment of a romantic emigre, one who came to London after Soviet tanks had crushed the Prague spring in 1968, with £50 in his pocket. He had worked in the Renzo Piano and Richard Rogers team that designed the Pompidou Centre in Paris, and when he met Levete had just been fired from Norman Foster's office because "he was too much of a maverick. I fell for all of that."

It's easy to overstate the couple's differences and their potential for creative symbiosis, to cast him as dour, masculine, iconoclast and tall, her as sunny, feminine, pragmatic, small – but there is something in that. She persuaded him to stop teaching and get an office where they could begin to build on the precedents established by Rogers and Foster, toward a more organic, voluptuous, formally inventive architecture. That office is the warehouse in Notting Hill where we're doing this interview.

It was here that the couple designed the Lord's media centre, an egg-like structure sheathed in aluminium panels. "That structure, probably more than any other, expresses the ideas, the aesthetic and the technical innovations that Jan had been exploring relentlessly for more than 20 years. That was also the year our son Josef was born – without question the best work we made together."

The £5.8m design almost bankrupted them, but when it won Britain's foremost architectural award, the Stirling prize, in 1999, the practice took off. But living and working together with no boundaries proved too much. "Ours was a very public falling out, played out in the office." They divorced in 2006, but carried on working in the same building. "For the last few years, there was a Berlin Wall between us in the office. Awful, awful, awful."

On 14 January 2009, Kaplický collapsed on a Prague street and died, aged 71. Hours before, he had visited his second wife Eliška and new-born daughter Johanna. Twelve days later Amanda met Eliska for the first time at Kaplický's funeral in Prague. "My greatest regret is that I didn't make peace with him in life," she said shortly after. A few months before his death, she and Kaplický had agreed he would move out of the office they had shared for 20 years, retaining Future Systems with a team of four, and she would remain in their Notting Hill warehouse as Amanda Levete Architects. "I'd hoped this would have made things easier. But we never found out if that would happen."

Levete is now married to Ben Evans, director of the London Design Festival. Amanda Levete Architects is thriving. Why are so few leading architects – you and Zaha Hadid notwithstanding – women? "Women leave to have babies and don't come back. It's a tough thing to be an architect. One of the hardest things for me is that I get described as super-tough. No man would ever be described that way – at least not as a criticism." Is it fair? "I think I'm a very benign boss. I'm also very demanding."

She shows me artists' impressions of her recent work, from a cultural centre in Lisbon to a tower block in Shoreditch. And then my favourite Amanda Levete scheme – a metro station in Naples designed with Anish Kapoor. Why couldn't they have done up my tube station, Finsbury Park? "Because there are very few visionary pieces of public patronage in Britain nowadays. Gove just expresses a more general contempt."

Shame. The design looks wonderful: one entrance looks like a rusting steel pair of lips, while the other is an aluminium form that seems to float in mid-air. It's great Levete and Kapoor will finally see a joint design realised. "Only one problem," says Levete. "For now, there are no other stations. We've designed a station for a subway line that goes nowhere." Hilarious, if a little embarrassing. No wonder some architects prefer their works to remain unbuilt.