Actually, the VP can't break a tie during contingent elections. There must be a majority.
Well actually, that's an untested constitutional question: the language requiring an absolute majority of Senate votes *may* preclude the sitting Vice President from breaking any tie which might occur, although some academics & journalists have speculated to the contrary, w/ legal scholars differing on whether the Constitution would allow the sitting VP to break the tie on the choice of a VP.
There was discussion before the 2016 election, that the senate may end up 50-50 and that there would be no electoral college winner (for example if Trump had done worse in the Rust Belt and McMuffin taken Utah).
That scenario would only be critical if the House failed to elect a President before 20th January. I suspect the Senate would delay electing a Vice President until it was known who would be the next President and would then agree to allow the election of the President-elect's running mate.
