Here’s how you handle this:

Announce that Mitch McConnell’s office has until the end of the week to respond in a satisfactory fashion to your entirely reasonable letter about his health. If they do not, at noon on Monday you’ll be holding a press conference on the front steps (or equivalent thereof; the hospital probably doesn’t have a lot of steps) of George Washington University Hospital in DC, which is supposedly where he is. After that press conference, you will be entering the hospital and demanding to see Senator McConnell.

If you are provided sufficient evidence that McConnell is alive and able to perform the duties of his office, you will come out and tell everyone that. If you, the governor of Kentucky, the state that McConnell represents, are denied the chance to see him, either by hospital staff, DC police, McConnell’s staff, or literally anyone else, you will be notifying the still-gathered news media of that fact and then you will immediately either appoint a new Senator in McConnell’s place or name the date of the special election, depending on which method your office has decided you’re able to utilize this week. I feel like this part keeps changing.

They will sue you, if you do that. An awesome thing about facing a lawsuit, though, is that they would have to prove he is alive in order to sue you for unlawfully replacing him. Which happens to solve the exact same problem that you already had, so it’s all good.

It’s time to make an ass of yourself, Governor. Do it.