My conclusion was that in such a scenario, Republican congressmen would pick the next President, which would certainly have been Bush in that case. To remind you, if there is no candidate with a majority of votes in the electoral college, the House of Representatives votes, each state casting a single vote, and a majority of all states is require. This has happened only once since the Twelfth Amendment was passed, in 1825.
In 2004, Republicans held the majority of Congressment from 30 out of 50 states, Democrats had a majority in 14, and six delegations were evenly split. By my calculation, a uniform swing giving Kerry a 7.2% lead would have been sufficient to get him the vote of 26 out of 50 state delegations; but of course it would not have mattered as by then he would have been far ahead in the electoral college anyway.
This year it's even tougher. By my count, Republicans currently have a majority of the representatives from 33 of the 50 states; Democrats again 14; and three are evenly balanced. Some of those we could naturally expect to shift back into the Democrat column in a half-decent year - the Republicans won the fifth of their nine seats in Arizona by 109,704 votes to 109,543 in 2014, so that will likely switch at the merest breath of a national swing. New Hampshire's two seats are currently split between the parties, but the Republicans will lose on a 2% swing. However as one goes down the list it becomes increasingly difficult to see where the Democratic gains come from. My rough calculations require a 7% swing to pull the Democrats ahead of the Republicans by 25 states to 24, with Nevada split evenly; the jump to the 26th state is then a good deal further. (See the very useful Daily Kos guide to the 2014 results.)
This only matters if there is no candidate with a majority in the electoral college. In 2004 (and 2008 and 2012) that seemed very improbable, requiring basically a tie at 269 votes each. This year things are different; might a mainstream Republican running against Trump as a third-party candidate, or a ragequitting Trump cheated of the nomination at the convention running against the official Republican, manage to split the vote three ways with Clinton (who to all but the most starry-eyed must surely be a cert as the Democratic nominee, but also clearly has difficulty reaching some parts of the electorate) to the extent that all three are deprived of the 270th electoral college vote?
Some are already convinced that this will happen. I can see a logic to it. For some political leaders, it may be worth having a messy convention outcome and a campaign riven by disputed legitimacy if the result is that the next President is chosen, not by the unreliable and fickle electorate, but by the disciplined members of the House of Representatives.
If so, the fate of the nation may end up resting on the decision of that third representative from Nevada.