How do I treat crumbling lath and plaster in my bathroom?
I recommend replacing all or part of the wall with drywall. Specifically, greenboard. If you patch only part of the wall you may need to use furring strips or use a thicker greenboard to make it flush.

As a bonus, if you rip apart part of the wall you may find there is something else hidden back there that is causing this issue which you can address at that time.