Welcome to History Hub,
I don't have your answer, but being a curious person, I am wondering why you think a President will not be chosen by January 20th?
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The place of the Speaker of the House in the presidential line of succession comes from 3 U.S. Code § 19.Vacancy in offices of both President and Vice President; officers eligible to act, not directly from the Constitution. Whether or not the scenario you are proposing would invoke this (much less whether such a scenario is even possible) is a matter for legal speculation rather than historical research.