Skyhawk, to answer your question I am going to say all of the above. In the situation at JFK and the one I had we were able to overpower the stuck mic to a degree the other aircraft could here our control instructions. The problem is that the aircraft with the stuck mic will never be able to hear anything over the radio. If you needed to communicate with that aircraft then visual means are all you have, for the most extent, such as light gun signals (which are only good if the aircraft is aware of the problem and cannot correct it and is looking at the tower). Or the runway lights which are not very noticable in the daytime and if they are it may take awhile for the pilots to realize what it means (wow....look at the runway lights....never seen them do that before). We can try to transmit over the VOR frequency (this will only work if the pilots have the nav radio volume up enough to hear which is not likely the case) I have sent out the Ops personnel to the aircraft to get their attention (this only works of course if the aircraft are on the ground....lol) The most successful means of preventing a stuck mic is the diligence of the pilots. Being able to realize that they have one in the first place is the biggest hurdle to overcome. Newer radios have an indicator that shows when the radio is transmitting, some have a time limit of how long they will stayed transmitting.
I'm not sure if there is a reg that states you will switch to 121.5 when a stuck mic is detected on the frequency but it is a good idea to monitor guard when able. I have used my ground frequency as a starting point to clear up a stuck mic issue but that would be difficult at places like JFK due to the amount of traffic on ground. But at smaller airports it's an option, but you have to be careful of airborn communications on a ground frequency due to the proximity of other airports that may have the same frequency for ground that would hear the towers tranmissions. This is the reason that airborn communications are restricted from ground control frequencies. My tower does not monitor 121.5 because we have a co-located FSS on the field and they are delegated to monitor guard (sounds silly dosen't it) We do have a portable transciever that we use in such situations if needed. Also, alot of general aviation aircraft only have one radio or only one that works....lol.
Sorry this ended up long winded but I hope I've answered your question. Remember I don't claim to know everthing, after 24 years as a controller I am still learning new ways to control traffic. There may be other opinions on how to handle these situations.