Senator William Blount of Tennessee was impeached by the House of Representatives in 1797. The Senate expelled Blount so the impeachment trial did not go forward.
If Boehner was impeached it would take every Republican in the Senate, including the RINOs to vote against him plus they would need 13 Democrats in order to get to the required 67 votes needed to remove him from office.
The original poster’s premise that it was a close election is flawed. Boehner received 216 votes to Pelosi’s 164 and 26 votes were cast for other candidates. The most any single alternative conservative candidate got was 12 votes for Representative Daniel Webster of Florida.
Learn something every day.
However, things were still shaking out in 1797, and it is at the very least questionable whether the House had the constitutional power to impeach Blount. The most commonly accepted opinion, with which I agree, is that the “civil officials” referenced by the Constitution as subject to impeachment are appointed by the President, not directly elected. (Or, in the case of Senators at the time, appointed by state legislatures.) So any executive branch official or judge can be impeached, but not reps or senators.
http://faculty.headroyce.org/~us_history/2004/c_kh/who.html
Since reps can be expelled by a 2/3 vote of the House, it would seem really odd to impeach by a simple majority of the House and then turn trial over to the Senate for a 2/3 majority. This would give the Senate power over the House, which the Constitution gives to the House itself.
The House can remove and replace its Speaker, anyway, by the same simple majority vote with which it could impeach him.
Impeachment of Boehner would also seem problematic because one would have to charge him with specific “high crimes and misdemeanours” rather than simple differences on policy. If Boehner has committed crimes of this type, so has probably at least 3/4 of the House.