What are ATC-side procedures to handle a jammed frequency?
Honestly, there is no guarenteed way we can solve such a problem, and it is extremely frustrating when it happens. A few possible solutions:

The radio transmitter associated with the tower frequency is usually quite powerful, so (at least for nearby aircraft) it may be possible to override the other transmission by simply transmitting normally. AFAIK, the transmitter is actually designed to be more powerful than aircraft transmitters as a safety feature. Sounds like this is what happened in your situation. This is usually only the case for tower frequencies (not ground/approach etc.)

Sometimes there is an obvious alternative frequency you can use, and maybe some of the pilots will try tuning to that frequency. We could also do a call on 121.5 and ask everyone tuned to the tower frequency to check for stuck mic. This wouldn't help for small GA aircraft with only one radio, since they won't be listening to 121.5.

In CPDLC airspace (en-route environment), ATC can send a "CHECK STUCK MIC" message to relevant aircraft.

In the end, the pilot with a stuck mic will usually realise after a while of apparent silence on the frequency.

Here is a recording of a stuck mic causing issues at JFK: