There would have been a special session to elect a Speaker of the House?
Otherwise, I've got nothing.
This is made somewhat more complicated by the fact that, constitutionally, only the president has the authority to call a special session of Congress. In any case, there still would have been a period of days or weeks where there was no president of the United States.
In practical terms, one can speculate the day-to-day work of the government would have been carried on by the cabinet, perhaps with Secretary of State Blaine as
de facto prime minister/acting president. It's not as if everyone would sit on their hands and do nothing due to a legislative oversight. Historically, presidents did sometimes leave their duties to the cabinet while away or traveling —Teddy Roosevelt famously disappeared into Yosemite for a month during his tenure and left the administration to Secretary of War Taft as informal "acting president," so there is some precedent for how this would have worked. My best guess is Blaine and the cabinet decide to bend the law a little bit and call a special session on their own authority, when the Senate would elect a ppt. Who that would be is anyone's guess; the Senate at the time was tied 37-37, so any candidate for ppt/acting president would have required at least one vote from across party lines. (Historically, independent David Davis was chosen as ppt and served until the end of the 47th Congress.) Of course it is possible the Republican House majority elects a speaker before the Senate can come to an agreement, which raises the question of "bumping." It would have been chaotic!
