Two Ks tell the story of Labour’s success in the general election – Kensington and Kirkcaldy. Two years ago, the party lost so badly that many wondered if there was any route back to power for it at all. Its support had retreated to English cities and Wales; it held only a single seat in Scotland and in the south-west of England. To form a government, Labour either needed to overturn the granite majorities of the SNP or to make gains in English seats it had rarely, if ever, held before.

On Thursday night, it did both. In Gordon Brown’s old seat of Kirkcaldy and Cowdenbeath, an SNP majority of 9,974 became a Labour lead of 259. In Kensington, west London, demographic changes and a strong preference for Remain dissolved the 7,361 majority of the Conservatives’ Victoria Borwick. Just after 9pm on Friday, a third recount delivered the seat to Labour by 20 votes. For weary activists, this was one of the sweetest victories of all: the constituency is home to the offices of the Daily Mail.

With results like these, something else dissolved in the early hours of Friday: any possibility of a leadership challenge to Jeremy Corbyn. This summer, there will be no need to argue whether seats or vote share is a better metric of success; the party improved on both. There will be no more talk of an existential crisis for the left; the electoral map now means it is entirely plausible for Labour to win the next election. And there will be no more yearning for a breakaway centrist party. Labour is the only game in town. And at its apex sits Jeremy Corbyn, who can lead the party for as long as he wants.

Labour’s right and centre must weather the gloating of Corbyn supporters who are demanding doubters eat humble pie

So what happens next? In the short term, Labour’s right and centre must weather the gloating of Corbyn’s supporters, who are loudly demanding that the doubters eat humble pie. The leadership is notably not joining in this kind of talk, because MPs and grassroots activists who have put their lives on hold for seven weeks to campaign for Labour don’t appreciate being treated like scabs.

Mike Gapes, MP for Ilford South, was one of the few to say publicly that the euphoria should be tempered by an acknowledgement that 262 seats gives Labour the power to do exactly nothing. “My constituents can’t live on humble pie,” he tweeted. “They need a Labour government. Now.” Others chafed at being called out of touch, noting that even Corbyn’s team did not expect this level of success. “The idea that John McDonnell and Jon Lansman [of Momentum] planned this is ridiculous,” said one insider. “They thought this was a defensive election – why else did Katy Clark and David Prescott [who work in the leaders’ office] demand safe seats?”

Nonetheless, almost all the party’s Corbynsceptics accept that they were wrong about Corbyn’s popular appeal. (Their concerns about his management style, ideology and past positions have not gone away.) The change in the tide was obvious when arch-Blairite Peter Mandelson went on television to recant. “An earthquake has happened in British politics and I did not foresee it,” he said, sackcloth and ashes just out of shot on College Green. Former leadership challenger Owen Smith echoed his contrition, as did Chuka Umunna, who also said he was ready to serve in the shadow cabinet.

Both Umunna and Yvette Cooper had been widely expected to challenge Corbyn after a wipeout. But both are now being urged by their allies to put their ambitions on ice. Clearly, the more voters saw of Corbyn, the more they liked him. He has also proved willing to compromise on key issues – shoot to kill, Nato, the monarchy, Trident – and the manifesto contained very little most members of the parliamentary Labour party (PLP) found hard to defend. “When tuition fees were introduced, the context was different,” says one MP. “House prices were relatively affordable and young people could buy within a few years of leaving university. But times have changed.”

Once Theresa May had set off for Downing Street, Labour MPs from all factions stayed quiet, giving the Conservatives a negative news cycle all to themselves. The prime minister’s embrace of the DUP might have solved her immediate problem of passing a Queen’s speech, but it has invited attacks over her competence and their social conservatism. (She really ought to have anticipated the trouble with leading a “coalition of chaos”.) The big Conservative winner of Thursday night, Ruth Davidson, has already flexed her newfound electoral muscle to criticise the DUP for repeatedly blocking attempts to legalise same-sex marriage in Northern Ireland.

Labour spirits were further raised by May’s tin-eared statement outside Downing Street, in which she did nothing to acknowledge the anger of her party for losing seats in an election she didn’t need to call. It seems there is no one in Number 10 willing or able to tell her that she often comes across as arrogant and complacent, another difference from Corbyn, whose advisers have successfully convinced him to hide his peevish irritation with impudent journalists.

For now, the PLP is waiting, because it feels Corbyn has to be the “first mover” in the reunification of the party’s warring factions. The first test is whether he will appoint any repentant Big Beasts to his shadow cabinet. It will be hard for Corbyn’s team to forgive their internal opponents – and to take jobs away from those who stayed loyal throughout the hard times – but accepting them back into the fold will smother their ability to agitate against the leadership. “If Dan Jarvis is on the frontbench, he can’t write articles about the McDonnell amendment,” is how one member of Labour’s right puts it, referring to plans to lower the nomination threshold in the next leadership election.

Unity will also be helped by the need to maintain a state of readiness for another election. Whatever Corbyn’s limitations as a manager and party leader, he has proved himself to be an incredible campaigner. (As glum centrists often observe: “He beat us twice.”) The Labour leader might not have taken his party to victory, but he has earned the right to fight again.

Helen Lewis is deputy editor of the New Statesman