Double sided tape..hmm... Either it will stick to the inside of the socket, or it´s not strong enough. Think of the force it takes to pull out a jackplug..all because of that little spring in the back..
Epoxy might be a good idea. It will give you 5-10 mins. before it cures. But it will feel like an eternity, if you are sitting and holding a stick in place .
Pull out your flashlight and check out the socket. Maybe you will get lucky with a little screwdriver. Have you tried to take a look inside the box? The socket is probably a closed unit, but if it has an opening in the back, it would make things easier.