But Brown, in his first comments on the plot, said: "It's not going to take up much of my time and hasn't, certainly, taken up much of my time.
"This is a bit of a storm in a teacup. We are actually dealing with real storms at the moment," he told BBC radio, referring to bitterly cold temperatures and snowfalls gripping the country.
In their letter to all Labour lawmakers, Hoon -- defence secretary when Britain joined the invasion of Iraq in 2003 -- and former health secretary Hewitt said the party was "deeply divided" over whether Brown should be leader.
"Many colleagues have expressed their frustration at the way in which this question is affecting our political performance," they wrote.
The letter gained early support from a handful of persistent Brown critics, but no major Labour figure backed the move, and Hoon admitted late Wednesday that it had failed.
Brown, who succeeded Tony Blair as prime minister in 2007, must face a general election by June at the latest. His Labour Party has long been more than 10 points behind in polls against the main opposition Conservatives.
Ministers rallied round Brown on Thursday after the failed leadership challenge.
"It is something we could have easily done without," Justice Secretary Jack Straw admitted, but insisted: "I think it will actually have the effect of bringing people behind the leadership even more."
In June last year, Brown fended off a nascent leadership plot after several government ministers quit in quick succession and one, James Purnell, urged him to step down.
Commentators agreed Thursday that Brown appeared to have seen off the latest attempt to unseat him, despite receiving only "lukewarm" public backing from some of his key Cabinet ministers.
Foreign Secretary David Miliband -- who reportedly considered standing against Brown in 2008 -- took seven hours to issue a statement which failed to mention Brown by name, commentators noted.
The Times newspaper described Miliband's statement as "less than wholehearted," while The Guardian daily said it was "both later and less full-throated than anything offered by fellow Cabinet members".
The BBC reported Wednesday that six Cabinet ministers had been prepared to support a coup. But Straw, among those named, on Thursday denounced the report as "sub-standard journalism".
Miliband insisted no senior ministers were involved in the plot.
"We have got an election to fight, Gordon is leading us into it, we are determined to win it, under his leadership, and I am looking forward to getting stuck into it," he told Sky News television.
Scottish Secretary Jim Murphy, also one of those named by the BBC, dismissed claims he was involved in the plot as "utter rubbish".
"It was a ludicrous thing to do, and the response from the vast majority of Labour MPs has confirmed that view," he wrote on his blog.
The timing of the row could hardly be worse for Brown: he and Conservative leader David Cameron effectively started election campaigning this week after the Christmas break, even though ballots are not expected until May.
Cameron said the fresh Labour infighting showed the government was "deeply divided".
"We cannot go on like this, we've got to have an election and a change of government," he said, adding: "If ever there was a time when our country needed strong leadership and united government, it is today."
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