The ranks of the Senate can be replenished with gubernatorial appointees very quickly. The Speaker of the House comes right after the Vice-President so the surviving representatives could elect a new Speaker who would then immediately be sworn into office as President and resign from Congress. Assuming “a Majority of each shall constitute a Quorum to do Business” in Article I, Section 5, Clause 1 refers only to members living & sworn into office and vacant seats don’t count then the solve surviving member would be able to elect himself Speaker and assume the Presidency.