You could try mounting it flush with the back edge of a dresser. A couple of vertical supports that can be screwed into the back of the dresser can also be attached to the back of the mirror. You could probably get away with a couple of 1" x 1" x 12" wooden boards that are screwed into the mirror box and the back of your dresser.
The whole thing can then be pushed up against a wall and you will have a mirror sitting on your dresser.
Edited to add: Or spring for a couple of flat aluminum stock. 1" wide, 12" long, 1/8" thick or so. Drill holes, screw into dresser & frame. This would add less space to the depth of your setup.
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