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You go to the parts store and get some rear view mirror adhesive and follow the directions. Don't take any shortcuts either or you will be doing it again soon.
how do you get the button off the back of the mirror so you can glue the button onto the windshield
I have an 89 and a 92 and one of those mirror plates you mention was metal and one was plastic. I think the adhesives to hold them on the window are different, as I recall. On these there is a pencil lead sized allen screw holding them to the mirror itself. Loosen that screw and it will slide out since you need to glue that little thing to the glass first without the weight of the mirror. Make sure you follow the previous poster's advice about surface prep and temperature of both the windshield AND the adhesive and mounting plates. Store the Glue inside over night and windshield in the sun. Razorblade the old glue off, super clean, maybe even alcohol rub, whatever they say on the directions. Also make sure the joints on the mirror itself are nice and loose so it doesn;t break off again. Guess how I know THAT little detail. Might even wait a day or so before you hang the mirror on it too. Amazing that it works but the glue is around 8 bucks so better not to have to do it more than once.