Within the tiny, interior of the Cuckoo Club in London, an eager audience has gathered -ostensibly to witness a performance from the jazzy newcomer Anjani. But everyone knows the real reason for their presence tonight. Leonard Cohen has been Anjani's significant other for eight years, and Anjani's new album, Blue Alert, is written and produced by him. He doesn't make many live appearances these days (this will be his first in the UK for 15 years), and never graces anything as trivial as a showcase, but for his woman he has made an exception.

A little after seven o'clock, Cohen, elegant in a light grey suit, walks onto the stage and clears his throat in a manner that conjures up images of rusty nails scattered across a metal floor. Within his shrewdly deferential introduction - he mentions that he understands that showcases like these can be a trial for the artist concerned, as journalists rarely shut up when chatting during songs remains a viable alternative - he effectively demands respectful silence, and promptly gets it. By the time Anjani - a handsome woman whose face is all cheekbones - takes the stage, you can hear a pin drop. For the next half-hour, she sings songs from her new album, each delivered in an imperious jazz style that brings to mind the cigarette-soaked films of Alain Delon. Cohen guests on "No One After You", and the audience watches agog, for this may be one of the last times they see the famously reclusive 73-year-old in the flesh. For his part, Cohen barely registers a smile, and his singing voice increasingly resembles his spoken one, but it's a magical thing, emanating from the very deepest of wells and dispatched with a minimum of effort but a maximum of gravitas. The Grim Reaper couldn't sound more foreboding.

Two days later, in one of London's more upmarket hotels, Anjani talks in a whispery voice: "I feel that the gods are finally behind me."

Blue Alert was released last year, but after a word-of-mouth campaign and a helping hand from the Canadian singer Madeleine Peyroux (who covered the track "Half the Perfect World"), it is now being re-released and re-promoted. "But you know what?", Anjani continues. "Its ultimate fate is completely out of my control, and that's just fine. Once I would have been anxious about it, I would have lost sleep. But I've reached a certain age now, and I've managed to see off all my demons. I'm not anxious any more."

She was born Anjani Thomas on 10 July 1959, in Honolulu. Her parents worked in local government, and she grew up with two brothers and one sister, in thrall to the music of Henry Mancini and Frank Sinatra. At the age of 13, she fell into what was only much later diagnosed as a clinical depression.

"Youth was a terrible time for me," she says now. "I had a chemical whatever in my brain that simply wouldn't allow me to be happy. Of course, I didn't know that at the time. All I knew was that I was in a completely foul mood, forever irritable." This cloud of depression would live with her for the next 20 years, throughout school and her formative years within a music industry that barely realised she existed. And, when she was 28, it saw off her first marriage, to a lawyer, after just 18 months. But then, five years later, it suddenly began to lift.

"I woke up one morning and just felt that things weren't so terrible any more," she notes, eyes wide at the recollection. "The day after that I felt even better, and so my life became bearable. It's quite an interesting revelation when that happens. I started to function like a normal human being. Not that I was a completely horrific mess before - I could laugh and have fun, though only intermittently. Mostly, my life was defined by sadness and dissatisfaction. All I wanted was to become enlightened, and was forever looking for true cosmic happiness. I never found it."

She first met Cohen in 1984, when she guested as a backing singer on his world tour. Thereafter, now settled in Los Angeles, she tried to establish herself as a solo jazz singer. "But nobody wanted to know." In the 16 years before she managed to record her first album, she sang jingles, played with fusion bands, and performed backing vocals for visiting musicians (among them the R&B singer Carl Anderson). Sometimes she'd make money playing LA's jazz circuit, often making not much more than $30 a session.

"After the big earthquake in 1993, I decided to leave LA, to break out, to try and live like a normal person." She moved to Austin, Texas, and got a job in a jewellery store. "I wanted to see what life was like on the other side, with health benefits, a stable job, a proper house. It was wonderful and horrible at the same time, simply because it was so against my nature." And then, at 39, she hit a crisis of sorts, convinced that if she didn't attempt to make an album now, she would regret it for the rest of her life. So she quit her job, sold her house and moved back to the West Coast, where she funded her eponymously titled album herself, released it in 2000, then watched it die. Shortly after, her path crossed once again with Cohen's.

"Our spiritual experiences have really paralleled each other's, both having suffered from depression and having it lift at the same time, and it sort of drew us back together," she says. "He was very encouraging about my musical career, and it was with his help that I began Sacred Names [her second album, released in 2001]." Encouraging he may have been, but in 2004, when she guested on his Dear Heather album, she learned that her paramour didn't necessarily pay compliments as easily as he did critiques. "I played him a particular vocal I'd recorded and I was convinced he would love it because I thought it was the vocal performance of my life," she says, quietly, "but he wasn't moved by it, and he told me so. Very plainly. In that instant, I felt that he had taken away all my confidence. I felt drained, exhausted, as if all the blood was running right out of me, and I started to shake. But it was also an epiphany of sorts, because, of course, he was right. We've worked well together ever since, because ours is a special sort of synthesis, it really is."

By the time they started collaborating on Blue Alert, Anjani going through Cohen's notebooks and adapting his words in a jazz style she feared he may loathe, she was convinced that she was merely creating demos that he would later record. But Cohen had other ideas. "He said that this should become my album, that my treatments had made them my songs alone. That's why its subsequent success has been such a surprise. It's really given me the impetus to run with it now. I've so much music inside me that needs to come out."

She says that the couple make music together to the exclusion of all other things. "We don't really have time for social pursuits. Life is so short, don't you find?"

'Blue Alert' is out on Sony BMG on 23 April