If a few of my assumptions here are met, you might try the following. I'm assuming that you are at a different institution now (though that matters less than) and it hasn't been so long ago that you finished the doctorate that you aren't still remembered at the old place.
Ask the Department Head at your degree granting institution to provide a letter that says that Dr. X is not currently capable of writing letters of recommendation due to medical issues that may or may not resolve positively. If the chair him/herself can attest to your good work, that can be added as well.
The Head may actually more comfortably correspond with your advisor to see if this, or some other, solution is possible.
The point is that the letter should come from someone official. It should give minimal details (privacy reasons), and should indicate that a letter may be forthcoming if possible.
There is one other thing the Head might also be able to do in a large department, depending on its organization. If there is a secretary who normally handles that professor's correspondence, he/she may be in regular contact and so a letter might actually be very possible. But again, if the request comes via an official request you can maintain your respectful distance while the advisor heals.