I've seen a couple of ideas that are intended to help resolve deadlock in the state legislatures.
One idea would be to have the state Senate and House meet as one body to cast a vote, in case the two houses deadlocked on picking somebody.
The other idea would simply be to have one Senator picked by each legislative house, thus sidestepping the issue of deadlock entirely, provided one house doesn't deadlock internally.
I suppose an absolutist’s view of “states rights”
would require the federal government to recognize
that each each state had the authority to determine
their own method of choosing their U.S. Senators.