It looks like the ball is in the sheriff's court. I found another authority on the governor's inability to remove a county elected official. Remember the county clerk that refused to issue marriage licenses to same sex couples a few years ago?
The wikipedia article about her cites a CNN story where it is stated that the governor has no legal authority to remove her or relieve her of her duties.
Kentucky law seems to treat most elected officials in the same way so I would think this restriction on the governor applies to sheriffs as well.
It would seem it would take an act by the General Assembly to impeach him and the grounds would have to be malfeasance, misfeasance, or willful neglect of his duties. There would likely also need to be an impeachment trial in the Senate. I would think he would have the right to defend himself in person which would be a major hassle. It's also not at all clear that what he did would even meet the bar for malfeasance or misfeasance. Misfeasance is civil abuse of power. Malfeasance is a criminal abuse of power. But he can just say he acted as himself when he shot the judge. He is unable to exercise his duties while in custody but that is hardly "willing."
I do think he should resign. But if he chooses NOT to, there seems to be very little anyone can do to force him out.
Oh, and by the way, the legislature is in adjounnment so it would require the governor to call a special session!
IMO, Kentucky needs to pass some constituional amendments to deal with the crazy world we live in!