His running mate would be sworn in as Vice President on March 4, then immediately sworn in again as President. The relevant text from Article II of the constitution:
In case of the removal of the President from office, or of his death, resignation, or inability to discharge the powers and duties of the said office, the same shall devolve on the Vice President
It only takes a slightly broad reading of this to apply it to your scenario: the elected President is dead, so the VP becomes President. Without the 20th Amendment, he needs to be sworn in as VP first rather than immediately becoming President-Elect in Knox's place. This reading is affirmed by the 12th Amendment's description for the procedure for a deadlocked contingent election for President:
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.
A strict reading of this suggests the VP-elect would become Acting President rather than President, but 1) that's a distinction without a difference when there's no chance the President would seek to reassert his office and there's no provisions for a special election, and 2) the Tyler precedent (that the VP becomes President, not Acting President) is very firmly established by this point.
In addition, there aren't any suggestions in the Constitutional text or historical precedent suggesting anyone other than the VP-elect might become President in this scenario, so there's no real endgame for anyone who wants to argue against the "VP-elect becomes President" interpretation.