Mr. Brownback began handing off tasks to Lt. Gov. Jeff Colyer after he was nominated, but month after awkward month passed in Topeka without a confirmation vote. Mr. Trump had to renominate Mr. Brownback this month after the full Senate declined to vote last year.

“Our state’s been in kind of a leadership vacuum of chaos,” said Jim Ward, the Democratic leader in the Kansas House of Representatives and a candidate for governor. “It’s really slowed down and pretty much stopped all progress on any kind of policy because you’ve got two different guys acting like the governor.”

Mr. Brownback announced on Thursday that he will resign as governor next week, on Wednesday. “Wherever my new duties may take me, my Kansas values and experience will always travel with me,” he wrote in his resignation letter.

Mr. Colyer, a Republican and a plastic surgeon from suburban Kansas City, will assume the governorship. Mr. Colyer, who is seeking a full term as governor in the election this year, will face several challengers in the Republican primary.