The House would have to continue to vote until somebody has a majority. If the House did not choose a president by January 20th then the vice-president chosen by the electoral college (or the senate) would become acting president until the House gets to a majority. So the VP would become president but it would not be a permanent position. Essentially what this means is that the executive power of the people must be vested in somebody, and it would be vested in the VP until a president would have been chosen. If the senate is likewise deadlocked in the the choice of VP, then the executive power would pass temporarily to the Speaker of the House, then the Senate President pro Tempore, etc, under the law.
The person having the greatest number of votes for President, shall be the President, if such number be a majority of the whole number of Electors appointed; and if no person have such majority, then from the persons having the highest numbers not exceeding three on the list of those voted for as President, the House of Representatives shall choose immediately, by ballot, the President. But in choosing the President, the votes shall be taken by states, the representation from each state having one vote; a quorum for this purpose shall consist of a member or members from two-thirds of the states, and a majority of all the states shall be necessary to a choice. And if the House of Representatives shall not choose a President whenever the right of choice shall devolve upon them, before the fourth day of March next following, then the Vice-President shall act as President, as in the case of the death or other constitutional disability of the President. - 12th Amendment
We have a massive advantage in States. The Dem’s EV’s come from a small number of states.