I don’t think there’s a clear answer, but I’d argue that we could conclude the balloting was over.
The point might be that the Senate can only vote on the people who actually ran for Vice President, and can only pick the top two vote-getters (the House when doing a contingent Presidential election picks from the top three finishers), so the new President in this weird scenario may pick a different person who might break the Senate deadlock, assuming it isn’t deadlocked based on purely party reasons.
What’s interesting is despite two amendments designed to clarify issues of Presidential election and succession there are still things left unclear. Article II, Section 1, Clause 6 of the constitution provides that when a Presidential vacancy occurs, the “powers and duties” of the office devolve to the Vice President “until the Disability be removed, or a President shall be elected.” When John Tyler became the first Vice President to succeed to the Presidency, he asserted that he was “now the President”, not the “Vice President acting as President.” Some of his political opponents disagreed–and said there was an argument to be made that Article II actually suggests Congress could call a special election to fill the vacancy (in fact the Presidential succession laws effected in 1792 allowed for a special election for President, but only in cases of both President and Vice President being vacant). But nothing ever came of it. The “Tyler precedent” was then followed when Zachary Taylor, Lincoln, Garfield, McKinley, Harding and Roosevelt died in office. The 25th at least clarified the matter by making it clear in its first section that when the President dies or is removed from office or resigns, the Vice President becomes President. But there’s actually still a potential for an ambiguous situation.
The 12th Amendment says specifically that in a case where the House cannot choose a President, but the Senate does select a Vice President, the Vice President elect becomes “acting President.” The 25th Amendment only specifies that “In case of the removal of the President from office or of his death or resignation, the Vice President shall become President.” So the 25th doesn’t actually contain any text that would counter the 12th Amendment, and thus the argument could be made that in this scenario the Vice President is only “acting President” and you could go back to Article II and make the argument that Congress has the power to call a special election.
There’s also arguments that the never-invoked 1947 statute on Presidential succession is unconstitutional on at least two grounds:
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The constitution provides that Congress can designate which “officer of the United States” shall act as President when both President and Vice President are incapacitated/dead/resigned, and technically members of Congress are not “officers of the United States”, a specific term of art in the Constitution referring to persons who have been selected to high government office by the President and confirmed by the Senate. Since this doesn’t apply to the Congressional leaders, they should not be in the line of succession.
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The constitution provides that when a person becomes “acting President” they are to remain “acting President” until another election is held to select a new President. The 1947 statute violates this in several ways. For one, it provides that in a case where President, Vice President, Speaker, and President Pro Tem of the Senate are vacant, the Secretary of State becomes acting President, but it also allows for that person to be “removed” as acting President whenever the Senate or House convenes and fill those vacant offices. That is in direct contradiction to the Article II requirement that a person who has become Acting President retains those powers until another President is elected. Likewise, combined with the provisions of the 25th Amendment, the 1947 statute would allow for an Acting President to nominate to Congress a candidate to fill the Vice Presidential vacancy–if that person was confirmed by both houses, that person would become Vice President–at which point you can argue he’d “bump” the acting President out of office (unconstitutional) and it would also put the new VP in the weird position again of being “acting President”, because the scenario under Section 1 of the 25th which specifies the situations in which the Vice President “becomes the President” aren’t necessarily met here.
[Interesting West Wing TV series sidenote–during the acting Presidency of Glenn Walken (Played by John Goodman) he says at one point that if his time as President continues for much longer he wants to “take action” on filling the Vice Presidential vacancy. Technically under the 1947 act, doing so would “bump” him, as someone who was acting President due to being Speaker of the House, out of office as acting President. Also in this story arc Walken resigns from the Speakership first, then takes the oath of office of President. Technically this could not be–the Speaker’s ability to be acting President is based on being the Speaker, in the moment he resigned his Speakership he was no longer legally able to be sworn in as acting President.]