Elly Jackson last released an album in 2009. Once, a pop star leaving five years between their debut and its follow-up would have been unimaginable – so much would have happened in the interim that it would be almost impossible to keep up, to seem like anything other than a relic from a lost past. Even today, when rock and pop develops at a far slower pace, five years seems a very long time to wait. In fact, it seems like prima facie evidence, as the website Popjustice sorrowfully concluded late last year, that "something has clearly gone very wrong", a judgment underlined by the curious, stuttering updates on the album's progress. In late 2009, it was announced that Jackson and her collaborator Ben Langmaid had started writing the album. A year later, it was suggested that she was now working with a mysterious "established indie band" – which could theoretically have meant anyone from Arctic Monkeys to Milky Wimpshake, but rumour insisted it referred to portentous doom-mongers White Lies. A year after that, she was said to be working with Nile Rodgers. Then in 2013, she played a series of low-key comeback gigs, followed by another 12 months of ominous silence.
As it turned out, almost everything seemed to have gone wrong with La Roux's second album. In the wake of her debut's success, Jackson suffered from anxiety attacks that eventually prevented her from singing. At one point, she thought she had throat cancer. Langmaid – who never appeared on stage, but co-wrote and co-produced every track on their eponymous debut – acrimoniously departed midway through the album's making. Clearly, there have been a lot of false starts: judging by the credits, whatever Jackson was working on with White Lies, Nile Rodgers, or indeed Milky Wimpshake, appears to have been left on the cutting-room floor.
It all sounds like a disaster waiting to be listened to, which makes actually listening to it a pretty startling experience. It's not just that Trouble in Paradise is a better album than her debut, although it is. That album boasted plenty of great songs, not least the chart-topping Bulletproof, but it also boasted a sound so obsessive in its recreation of the early 80s as to border on the parodic. There were moments when La Roux sounded worryingly like a Not the Nine O'Clock News sketch about synthpop: furthermore, the tinny production was thrown into stark relief by remixes like Skream's version of In for the Kill, which sounded as appealingly forward-thinking as the music Jackson was pastiching had done 25 years before. On Trouble in Paradise, however, La Roux's sound has expanded its horizons. That audible love of the 80s is still present, but the synths are bolstered by scratchy funk guitar, the steady pulse of the dance music that held sway in New York's gay clubs between the decline of disco and the rise of house, and luscious pop textures drawn from a couple of the decade's more off-kilter hits. If Nile Rogers himself isn't present, the influence of Carly Simon's astonishing, Chic-produced 1982 hit Why still hangs heavy over Tropical Chancer (the frantic, angular guitar solo at the end of Let Me Down Gently, meanwhile, sounds like a nod to the sound conjured up by Rodgers for David Bowie's Let's Dance). Elsewhere you can hear the influence of Grace Jones' stately epic Slave to the Rhythm. Jackson's voice – previously striking in a way that was rather a mixed blessing – now feels far less shrill. The whole thing seems lithe and supple, where its predecessor was willfully stiff: you're no longer haunted by the sensation that Mel Smith is about to appear in lipstick and eyeliner, causing the audience to bust a gut.
But the real surprise is how unforced it all feels. When an album has suffered a tortuous gestation, and when it's been labored over as intensely as Trouble in Paradise has apparently been – "14 to 16 hours, six days a week, sometimes longer, for two years," Jackson recently told the Observer – you can usually tell just by listening to it. But if someone didn't inform you of Trouble in Paradise's backstory, you'd never guess. The only hint of hard work comes with the track listing's labored puns (Tropical Chancer can take its place alongside Kiss and Not Tell and Sexotheque, every one of them a song as fantastic as its title is rotten), while the only suggestion of discord is in the lyrics of the lengthy Silent Partner, which appear to address both Jackson's illness and Langmaid's departure, the latter in pretty withering terms. It sounds airy and confident and effortless, a state of affairs aided by the fact that, at some point in the last five years, Jackson and her various collaborators have hit on the knack of coming up with songs that somehow sound as if they've always existed, as if you've stumbled on a selection of old hits that you'd forgotten about but are delighted to be reminded of. Conjuring up that weird, false sense of instant familiarity is one of the most potent and difficult tricks in pop music. It's what lies behind the mammoth success of both Daft Punk's Get Lucky and Pharrell Williams's Happy, and it happens over and over again on Trouble in Paradise, most arrestingly on the opening trio of songs: the single Uptight Downtown, the Abbaesque Kiss and Not Tell and Cruel Sexuality, which it seems fairly safe to say, is the most sublimely euphoric exploration in recent pop history of the pressures placed by society on the individual who declines to define themselves as either straight or gay.
In truth, the songwriting quality never really dips. Almost sickeningly overburdened with fantastic tunes, Trouble in Paradise may well be not just a triumph against the odds, but the best pop album we'll hear this year. Listening to it, it's hard not to feel that whatever agonies went into its creation were worth it.
• Alexis's original score was amended on 18 July from 4 stars to 5 stars
