The SIMPLE fix is to pull the mirror off carefully with your hands and clean the back of the mirror and wall behind it with 91% alcohol to remove ALL of the old double sided tape and residue. Go to WalMart or an auto parts store and buy some wide 3M automotive gray double-side tape and place several strips horizontally along the back of the mirror a few inches apart.

Now, here's the key: put large blots of clear RTV on the back of the mirror BETWEEN the double sided tape strips. THIS is actually what will REALLY hold your mirror to the wall. Use a goodly amount, but not too much that it interferes with the double sided tape. Make sure you use clear RTV; it is removable by rolling it with your thumb should you ever want to remove it and it will NOT hurt the surface it is applied to.

Hang the mirror back on the wall.
The double sided tape's primary purpose is to hold the mirror in place while the RTV cures (about 24 hours) and even if the double sided tape "lets go" (highly unlikely with the 3M product) the RTV absolutely WILL keep the mirror from falling in the future.
Again, the 3M automotive double sided tape is GREAT stuff and won't likely fail especially if you use enough of it and prep the surface with the 91% alcohol. It is designed to be used on shiny automotive exterior paint in the blinding hot sun while exposed to the rain and elements to hold trim and molding on cars. Sure, it's hot in an RV when the A/C is off stored somewhere when not in use, but not NEARLY as hot as on the bodyside cladding of a 1995 black Pontiac sitting in the blistering Arizona sun...which is what the 3M tape was designed to be used on.
However, in the event that the 3M tape DID fail, the RTV again WILL keep your mirror from EVER falling off again. Both of them together are bullet-proof. That laminate wall is going to come apart before the 3M tape and RTV fails.
This is an EASY fix that WILL work...and you don't need to use ugly dollar store 1950's mobile home mirror clips or do a patch job by adding out of place wood strips to hold the mirror in place.
Hope this helps!