While the Constitution doesn't make it absolutely clear, presumably they would use the precedent from the 1800 election. (The 1824 election produced no tied State delegations, and 1876 was a dispute over which electoral votes were valid.) While the Twelfth Amendment did make some changes in how the election would be conducted, it left unaltered the method of election in the House should it be up to it to decide, so the precedent should hold. For 35 ballots, the result was 8 states Jefferson, 6 states Burr, 2 states tied and no result. Thus to be elected in the House, you need a majority of the Representatives of a majority of States. Assuming a straight party line vote, the current House would go 30-17-3 for the GOP.
Because the Democratic strength in the House comes largely fom dominating the California and New York delegations by large margins while the GOP controls most of its delegations by slender margins, the Dems fare poorly in an electoral vote tie.
I wouldn't assume a party line vote. I think many represenatives would either voe based on how their district or more likely how their state voted. This would still favor the Republicans though since they dominate the small western states.
