Frank LaRose might run for Secretary of State. He would do pretty good.
This alone shows he would do a pretty good job, IMO. Other than that, he's proposed legislation introducing online registration to vote and making the absentee voting system more fair and accurate. Other than that, "[h]e sponsored legislation to make Ohio’s regulatory system more efficient and more conducive to economic growth," according to Wikipedia, so he could try for Treasurer.
Mike Duffey was Kasich's intern in D. C. and worked for Kenneth Blackwell as Secretary of State, so he's another possibility for Secretary of State. Duffey also worked at a public relations firm. In the legislature, he helped privatize the Department of Development's oversight, allowing it to be more efficiently ran.
Pat DeWine, meanwhile, is far and away the most likely/qualified to be Attorney General next. I'd be very surprised if he doesn't run.