If he'd put through a deal that involved abandoning the Tea Party and bringing in some Democrats he'd have lost the long game. As well as likely lost his position as speaker.
Right now one of the major things that gives the Republicans their power is their party unity. They are working more like a party in a Parliamentary system, voting as a bloc, etc.
One thing the Democrats have failed to understand is that the Republicans are very good at playing the long game and it's pretty much all they do. Losing a vote here and there, losing an election here and there, being blamed for failures by the President, is all stuff they'll pay as a price for pushing the long term politics of the nation more their direction. Why do you think that the Democratic party in the USA is generally to the right of the right wing parties of Europe? It's because the Republicans are very good at moving the Overton Window.
That's why they've got a whole branch of people devoted to being publicly seen as crazy who have the job of pushing really out on the fringe ideas into the mainstream. The militia leaders, backwater talk radio hosts, less respected weekly newspaper columnists, etc all take ideas that are currently so far out of the mainstream that no one who pushes them can be taken seriously, and they push them at the cost of their own credibility. Then the higher tier commentators talk about the "controversy" which moves the idea further into the mainstream. Repeat a few times and soon torture, a tax rate for the wealthy that's at a historic low, indefinite detention, public health care as Communistic evil, assassination programs targeting US citizens, endless decade long wars on three fronts, etc are positions that everyone including the Democrats take as given.
So no, Boehner will not break his party's unity to give the President a deal. In the short term it'd be a good thing for Boehner and the Republicans, Obama has indicated that he'll trade away massive cuts in Social Security, Medicare/Medicaid, etc for pocket change level (and temporary) tax increases on a few millionaires. If the Republicans were playing the short game they'd take it and count it as a win.
But they aren't. If they crater the economy, or incur some short term cuts in military spending, that's not good from their POV, but in the long run maintaining their party discipline is more important.