Wouldn't she just appoint a Democrat to her seat? Doesn't matter, though, she only hangs on if a Republican is President in 2018, most likely. North Dakota would be the first seat to fall in a midterm election. Even with a GOP President, it's still possible that it could fall.
A governor-elect can not appoint a senator, and a governor-elect can not become governor until (and if) they resign their other offices. If she resigned her senate seat, then the current governor could appoint a replacement.
At best she could hold out from resigning her senate seat, past the end of the gubernatorial term, but then you are in the land of fighting suits about what happens when there is no governor. Are North Dakota lieutenant governors elected independently of the governor?
