Maybe try double sided tape.
It is hard to know just what is needed here without knowing if the mirrors are framed, or frameless. With frames I would measure the exact length and width of each mirror. Next would be the selection of the fabric. Cut your fabric to exact length and width. Place double-sided tape on the mirrors all the way around the edge up next to the frame and then carefully affix the fabric to the tape, being careful to smooth out wrinkles as you progress. The next step would be to get some decorative edging material to go around the edge of the fabric, the point here is to cover the raw edges of the fabric. This can be glued to the fabric with fabric glue, purchased at any fabric store, probably where you purchase your fabric and edging material. Or you could use a second application of double sided tape.
Anyone who works in a fabric store can fill you in on details of just what you will need. Without seeing the mirrors I would hesitate to give you suggestions for covering unframed mirrors. Maybe talking with the person in the fabric store will be helpful to you.
I have heard of military wives using liquid starch on fabric and smoothing it on, then squeegeeing it out as it's applied to the wall. That way, they just pull it off the wall when they are transferred to another base housing area and wipe the wall down to get the dried starch off. From what I understand, this was a common practice. I don't know why it wouldn't work with a mirror. Good luck!
A spray adhesive may work well, but why not just remove the doors and put in curtains instead?