For the hundredth time:
- When the House elects a President after an Electoral College deadlock, every state gets one (1) vote. California gets one vote, Wyoming gets one vote. Population, or number of representatives, doesn't go into it.
- #NeverTrump Republicans are a very small minority of the caucus. However, they are disproportionately located in small states, like Alaska, Utah, Idaho, and Nebraska. For this reason, Donald Trump cannot win a House ballot except in the event that the House GOP gains seats outright, and in this event he's probably already won with >270. I am willing to go into much greater detail about which individual members are likely to lose or likely to oppose Trump, and there's a good deal of whiffle either way, but it's very likely that Trump finishes with 20-24 delegations; short of the 26 he needs to win.
- An anti-Trump Republican candidate -- whether it is Gary Johnson, Evan McMullin, or somebody McMullin drafts to contest the House election, as Fuzzy Bear speculates -- cannot possibly hope to win; too many Republicans are pro-Trump (a large majority of the caucus). Hillary Clinton cannot hope to win either.
- In the event Trump actually recovers enough to throw the race to the House, it is overwhelmingly likely that Republicans hold control of the Senate. In this case, while the top three finishers from the presidential race go to the House, only the top two go to the Senate (so, Mike Pence v. Tim Kaine; Mindy Finn would not be an option). Pence would therefore be the winner.
- If the House simply lets the position of President go unfilled, the Vice President (so, Mike Pence) would then become Acting President. Note that the House must vote on the position of President "immediately"; this probably means that they can no longer elect a President after they go on their first recess after the election and the Presidency is recognized as vacant, so the Vice President accedes to the post by default. Pence would become President in this scenario even if he actively opposes it and lobbies House members to elect Trump.
Why wouldn't the House, as a whole, go rogue?
Trump isn't anyone's first choice for President, and all established factions of the GOP are not happy with Trump's hostile takeover of their Presidential Nominating Process. There's no love lost, and Trump has given his GOP enemies all the excuses a politician would need to stab someone in the back.