I think that if you drilled a hole to install a plug, you would only want the plug at the bottom surface of the key. Only the bottom 1/8" or 4mm grips the balance rail pin closely. (Well, it doesn't grip, it is a loose fit, but it is tighter than the big slot in the key and the button.) The slot at the top of the key button is 7/16" from front to back. This oblong slot is 7/16" all the way down to the bottom 1/8" of the key.
Can you show us some photos of your keys? Show us the round hole at the bottom of the key, and the slot at the top of the key as well. If the hole at the bottom of the key has become enlarged, then the only reasonable explanation is that some insane owner thought that jamming needle-nose pliers in there to stretch the hole would give them some lighter or looser action. This person should be hunted down and spanked.
If you have to plug these keys to make the repair, you will have to use a piece of wood that is large enough to provide enough surface to glue and be strong enough to survive the drilling and years of playing without cracking or popping loose. Maybe test five keys at the treble end, and see how hateful the operation is. If it becomes too much work, bite the bullet and buy a whole new set of keys? Ouch.
I think that sawing a notch across the bottom of the keys will be easier than trying to drill out a hole for a round plug.
Sean