At the 2014 midterm elections, the Republicans achieved their largest majority in the House of Representatives since the 1920s. This resounding victory, party leaders boasted, would showcase their ability to govern. Instead, the debacle over the election of a new Speaker has demonstrated that Republicans are unable to govern themselves, let alone the country.

The reason for Thursday’s shock withdrawal of Kevin McCarthy from a contest it was assumed he would win was the same one that forced a battered John Boehner to resign as Speaker last month: the scorched earth hostility of a few dozen hardline conservative members demanding a greater say in running House business and an end to all compromise between the leadership and the Obama White House.

The deadlock threatens to paralyse Congress as crucial decisions loom, on an increase in the federal debt ceiling to avert a government default, perhaps as early as 5 November, and on a deal to prevent another government shutdown when the current stop-gap funding expires on 11 December.

On Friday, Republicans on Capitol Hill held frantic meetings. The focus is now on persuading Paul Ryan, the party’s 2012 vice-presidential candidate and chairman of the tax-writing House Ways and Means Committee, to pick up the poisoned chalice, with senior Republicans saying he is the only one capable of uniting the dysfunctional congressional party.

Mr Ryan has all along maintained he doesn’t want the job. But the pressure from colleagues may prove impossible to resist, and this weekend he is back in his Wisconsin district to discuss his options with his family. But top Republicans sound increasingly optimistic he will have a change of heart. “Paul’s looking at it. If he decides to do it, he’d be an amazing Speaker,” said Mr McCarthy.

Failing Mr Ryan, the likelihood is of continuing turmoil. Neither of the two remaining declared candidates – Utah Congressman Jason Chaffetz and the little known Daniel Webster who is backed by the conservatives – generates much enthusiasm. Other runners have yet to come forward, all waiting for Mr Ryan to make up his mind.

A large part of the problem lies in congressional arithmetic. A Speaker is elected by the whole House, Democrats included, and needs a majority of 218 of the 435-member body. But Democrats will almost certainly vote for their own leader, the former Speaker Nancy Pelosi. Thus although Republicans have 247 seats, a candidate can only afford to lose 29 of them if he is to clear the 218-vote barrier.

This means the 30 to 40 members of the Freedom Caucus, the hardliners’ group, hold a de facto veto. When the Caucus announced it was backing Mr Webster, the game was effectively up for Mr McCarthy – barring concessions to the conservatives that would have split the congressional party. “A banana republic” was how the moderate Republican Representative Peter King described the situation.

Also fuelling the disorder is the 2016 presidential race where the current top three Republican candidates (businessman Donald Trump, retired neurosurgeon Ben Carson and Carly Fiorina, the former Hewlett Packard CEO) are scoring a combined 50 per cent in the polls, despite none of them ever having won elective office.

Uniting this trio and the House conservative faction is the belief that Washington doesn’t work, and that existing Republican leaders have failed the party by not pushing through Republican policies.

“Kevin McCarthy’s out, you know that, right?” Mr Trump bragged to a rally in Las Vegas on Thursday.

“They’re giving me a lot of credit for that because I said you really need some-one very, very tough – and smart.”