Liz Kendall could be prime minister in five years, which is remarkable for someone few people would have known from a bar of soap before this month. She could win the Labour leadership in September, and then she could face – because David Cameron will be off – Boris Johnson, George Osborne, Sajid Javid or Nicky Morgan at the next election. Strange business, politics.

But first, how does she get to be Labour leader? Surely she is too little known, too right-wing, too “Blairite”, to have a chance against Andy Burnham and Yvette Cooper? Not at all. It is often forgotten that Labour Party members voted for David Miliband last time, rather than for his brother. Of the three sections of the electoral college which elected Ed Miliband, David did best, with 54.4 per cent of the vote, among party members. This time, thanks to the brave and right change brought in by Ed, this section of the old college will be the only one that elects the leader.

Of course, the party membership has changed over the past five years. People who joined the party under Ed’s leadership are likely to be more in sympathy with his views than older members. The other complication is that there are two groups of people who are also allowed to vote for leader on the same terms as individual members. Affiliated supporters, who are members of Labour-supporting trade unions, and registered supporters, which is anyone who signs up at the party’s website for £3. No one knows how many there will be in either category. No doubt the sub-Marxist cliques that run the big unions will try to organise their members to sign up and to vote for their favoured candidate, Burnham. But they won’t be able to include “Vote Andy” leaflets with the ballot papers this time, because the Labour Party will run the ballot.

And defeat brings its own discipline, especially when it is both emphatic and unexpected. Labour members want to win, and will be looking for the candidate most likely to beat the Conservatives.

One thing they will ask is, Who do the Tories want us to choose? It is a clarifying question, and we already know the answer. They want Labour to choose Burnham. If they can’t have him they want Cooper, although they realise that she is tough. But the one they don’t want is Kendall.

Not that they put it quite like that. One of Cameron’s advisers told me they were unimpressed with someone who had made so little impression since she became an MP five years ago. But Kendall’s obscurity is a strength. She is the candidate who can best refresh Labour’s image, and who can break most convincingly with the perceptions that put people off Labour.

Who will be the next Labour leader? Show all 7 1 /7 Who will be the next Labour leader? Who will be the next Labour leader? Andy Burnham Andy Burnham has promised to restore the party's "emotional connection with millions of people," if elected Getty Images Who will be the next Labour leader? Mary Creagh Mary Creagh has called on her party to win back “Middle England” Getty Images Who will be the next Labour leader? Liz Kendall Shadow health minister Liz Kendall is seen as a Blairite PA Who will be the next Labour leader? Yvette Cooper Shadow home secretary Yvette Cooper became the fourth person to join the Labour leadership race AFP/Getty Who will be the next Labour leader? Tristram Hunt Tristram Hunt, the shadow Education Secretary, has said he will not run for the Labour leadership as he had not gathered the required nominations of 35 MPs. He has instead endorsed the moderniser Liz Kendall. Getty Images Who will be the next Labour leader? Dan Jarvis One of the favourites to succeed Ed Miliband as Labour leader – ex-Army paratrooper Dan Jarvis – has ruled himself out, saying he won't do it because of his children Getty Who will be the next Labour leader? Chuka Umunna Chuka Umunna dropped out of the Labour leadership contest just three days after he announced he was in the running YouTube

The “Who do the Tories want?” question also helps to clarify the delusions of the pseudo-left, such as Zoe Williams of The Guardian, who said last week: “I actually have nothing against LK, she’s just in the wrong party.” Most Labour members realise that this is an observation in LK’s favour. She is a candidate who could persuade some of the people who voted Tory this month that they should vote Labour instead.

That is why I think Burnham, the early favourite, may turn out to be the David Davis of this contest. Like Davis in 2005, he has picked up a lot of nominations from MPs and party grandees such as Charles Falconer, but could be overtaken by a candidate better placed to change the party’s image who is a more fluent performer on TV.

Kendall sealed her advantage as a communicator at a press lunch last week. The Westminster media pack is a hard audience for any politician, but she impressed them with her directness. It may be that, ideologically, they (or I should say we) may prefer a centrist, but I think it was Kendall’s conviction that was more important.

Meanwhile Cooper is in danger of being left behind. She could have won the leadership last time, but she stood aside to let Ed Balls run. This time, she positioned herself in a Huffington Post article yesterday as the middle candidate between Burnham on the left and Kendall on the right, but she is already in third place in a (self-selecting) survey of readers of the Labour List activists’ website. If Kendall stumbles, and joins the discard pile of Dan Jarvis, Chuka Umunna, Tristram Hunt and, soon enough, Mary Creagh, then Cooper would probably prevail.