In Bangkok, is there a place you can stay privately to recover from an operation after the hospital releases you? Like a sort of hotel but with medical staff on hand?

Believe most nursing homes offer such.

 

What hospital? Most will let you stay on for extra days and do not charge that much. 

 

Several reasonably prices hotels about the perimeter of Bumrungrad that are an easy walk. 

 

 

I'd say ask the hospital.

It woukd be unusual gor a Thai hospital -- a private one at least --  to discharge you still needing such care. They tend to keep patients much longer than in the West.

 

What type of surgery will this be?

If you have the budget for it, Bumrungrad have a Rehab centre in Bang Krachao Bangkok.