Badenoch will be looking over her shoulder after less than emphatic leadership victory

The first black woman to be elected leader of the Conservatives and the fourth female leader of the party, Kemi Badenoch's no-nonsense style of politics in the end won out.

Party members and MPs took the calculation that Badenoch cut through in a way rival Robert Jenrick could not and be best place to take the fight to Labour. She has a hell of a job on her hands.

For a start, her win was not emphatic. It was the closest result of any Tory leadership race ever, as she clocked up 56.5% of the vote against Jenrick's 43.5%. Turnout was 72.8% of the party's 130,000 members.

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Badenoch won on fewer votes than Rishi Sunak lost to Liz Truss, who got 57% of the vote. This matters because it highlights the job Badenoch has on her hands to unify the party and ignite the grassroots.

Thousands of members were no doubt fed up that the final choice was two candidates on the right of the party, after James Cleverly's shock exit.

Meanwhile, in the parliamentary party, Badenoch has been a rather divisive figure, as MPs split into a camp that believes she is the real deal and others who find her abrasive and pugnacious.

She only won the support of a third of MPs in the parliamentary stages of this contest - as did Jenrick and Cleverly. From the party membership to MPs, she has a lot of work to do to build support.

"I personally think she will have a difficult time trying to form a shadow cabinet that reaches beyond her supporters," said one Tory source after the result. "I wouldn't be surprised if at least couple of candidates don't choose to serve."

There are questions about whether those on the right of the party, who had tried to oust Rishi Sunak months before he called the election and who, in the end, coalesced around the Jenrick campaign, will fall into line and support Badenoch.

Keen Westminster watchers will have noted how Sir John Hayes, a Jenrick backer and influential right-winger who heads up the Common Sense Group, told Sky News after the vote colleagues should get behind Badenoch "for the time being".

Another Conservative source in that grouping told me after the result that they recognised "a period of silence is required from us now" while adding that their "honest gut instinct, is that this is not going to work".

That was rather at odds with what Vicky Atkins, dispatched by Jenrick to mop up after the result, told me as she urged the party to unite behind Badenoch. I suspect that is what the party will now do, but even some of Badenoch's backers acknowledge privately that it's a gamble.

"She's a high risk, high reward outcome," said one of her supporters. "Bluntly, it wasn't a vintage year, but she was the best of the six," said another senior Conservative MP.

Beyond the politics of trying to unify the Conservative party from grassroots to parliament, Badenoch has a policy platform to put together. During the campaign, she didn't offer policy detail and said that would be her work should she win.

Her pitch to the party was that she was someone who spoke the truth and would reset the Conservatives' thinking, while arguing that her engineering background made her well-placed to fix Britain's "broken system".

David Davis, former cabinet minister and a Badenoch backer, told me that he thought she was the best equipped of the six runners to take the fight to Labour and build the policies to answer the problems "which Labour can't but which matter to the public".

"I'm weary of over-egging the comparison with Margaret Thatcher. But the crisis now in global and national terms is akin to what Thatcher faced when she came in in 1975 and the Soviet Union was in the ascendency, the US had lost the Vietnam war, the trade unions were dominant, inflation was out of control and unemployment was out of control.

"Today we are at war in Europe, there is massive instability in the Middle East, the NHS isn't working, the welfare system is paying people to sit at home. Housing is a disaster for young people and social mobility is at its worst in decades.

"Thatcher was a scientist and a lawyer who tried to solve problems by breaking them down into small pieces. Badenoch is also a lawyer and engineer which is a great combination for solving problems."

There are some, such as my fellow Electoral Dysfunction podcaster Ruth Davidson, that doesn't think Badenoch will necessarily even lead the party into the next election, on the basis that neither of the candidates (Davidson had backed Cleverly) has the breadth of appeal to push beyond the narrow base of the Conservative core vote.

And the scale of the rebuild is enormous. The July election was the party's worst-ever parliamentary defeat. It lost 251 seats and was reduced to just 121 MPs. Reform picked up four million votes, helping to drive the Conservatives out of dozens of seats, and the Lib Dems won a record 72 seats as they wiped out Tories in the South West and traditional shires.

But where the Tories do see a glimmer of hope is difficulties besetting the Labour government, in the wake of a messy first 100 days and a massive Budget that sprung on the public huge tax rises and spending plans that hadn't been spelt out in the manifesto.

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One poll put the Tories ahead of the Labour party for the first time in three years by just one point. It was only one poll, but it is giving the Conservatives hope that they might not be shut out of power for a decade or more if Labour continue in the manner by which they have begun.

Badenoch's first task will be to steady the ship. In the next 48 hours, she will pull together her shadow cabinet to present to the public a fresh Conservative party.

Then the grind really begins to win back the chance to be heard by an electorate that so roundly rejected the party. It's going to require a lot of work from think tanks and it's not yet clear what coalition of voters Badenoch is going to target to try to rebuild the Tory brand.

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But the lingering question is whether, as some MPs privately warn, Badenoch can survive to the next election given that she won the support of only a third of Conservative MPs in the leadership race.

One former cabinet minister tells me that her manner "won't endear her to donors, who are vital in opposition, while also managing the wide range of opinion in the parliamentary party".

This week the 1922 committee of backbench MPs changed the rules to make it harder for MPs to trigger a ballot - now a third of MPs must submit a letter of no confidence as opposed to 15% - in order to better protect the newly anointed leader from a challenge.

That they felt the need to do so is perhaps a nod to trouble ahead in a party still divided after defeat.