Presumably, the Senate would know in advance who the House will elect as President, so if Biden is going to be elected by the House, the Senate would probably elect Trump as VP.

Given Biden's mental state, he is likely to be much more than a figurehead President, and will probably not be able to serve his entire term as just a figurehead.

If Biden is President and Trump is VP, you may actually see democrats calling for Biden to either be impeached or to step down so Trump would not be eligible for a 3rd term if he serves less than 2 years as actual President after the 2020 election.

If you think politics were filled with drama today....