Are there procedures for pilots to safely handle a situation in which ATC loses the ability to communicate (complete loss of power, incapacitation of all controllers, etc.) at a busy towered airport?
Yes. Regardless of the source of the communication loss, much of the procedural framework in IFR, detailed clearances and flight plans, standard profiles for this and for that, Standard Instrument Departure procedures, Standard Terminal Arrival procedures, are to make it safe to complete a trip after a communication loss.
While in communication with ATC, you just do what you're told by the controller, and technically, none of those published procedures and protocols are really necessary when you just have to follow instructions.
When a comm failure happens, you notify ATC by selecting the Comms Failure Code 7600 on the transponder. The standardized procedures, and your clearance, gives you a path to take, and which ATC can assume you'll take to complete your trip.
When there isn't a specific published routing to follow, you are expected to find your way there by the most logical path so the controller can anticipate what you're going to do. For example, if you're on radar vectors to an approach and there is a comm failure, and you select 7600, the controller can anticipate that you'll navigate to the final approach path by the most obvious route and carry on with the approach and landing, and will make sure other airplanes stay out of your way.
If the comm failure was at the other end, at a control tower, the pilot would try another frequency before assuming there was a comm failure in the plane. For example, if handed off to tower by the arrival controller, usually when established on final, and tower didn't respond when checking in, the pilot would switch back to the arrival controller's frequency and let them know what's going on or check that they got the tower frequency correct.
However, in that case, it's likely that the arrival controller, who is at another location offsite, already knows about whatever crisis has befallen the tower, and ATC will be sending airplanes elsewhere anyway.