What would happen if..

[© tweed]

The House votes until it has a resolution.  Jefferson won on the 36th ballot in 1800. 

And yes, the sitting VP would break the tie as always in the senate.

[© tweed]

a quorum for the purpose shall consist of two-thirds of the whole number of Senators, and a majority of the whole number shall be necessary to a choice.

I'm not going to lie, I don't fully understand that and I doubt many people do.  It could mean 3 things,

-you only need 2/3 of senators to form a conclusion
-you need a majority of all sitting senators to form a conclusion (what you said)
-could that majority be 51/101 including a VP?

I wish they wrote these things so I could understand them.

[© tweed]

Given the absence of a tie-breaker when the House elects the President, I see reason to assume that they's have one for when the Senate elects the Veep.

But the house doesn't have a tiebreaker on anything else either.  The Senate has one built-in so the situation might be different.