I would mitre the ends and glue them with wood glue first. Then on the back I would tack lath to create a lip to provide an edge for the construction adhesive.
Before the construction adhesive (Liquid Nails) I would rough up the edge with emery cloth.
Baseboard would be fine....mitre the corners and then attach with glue or nails. We used picture framing for our big bathroom mirror. Picture framing has a lip on the back and you can nail the outer part of the frame to the wall.
Hi ! Pick out the trim you want , measure the mirror , cut the trim to size ( using a miter box $12 at home depot ) makes the cuts fit well . Then paint or stain it .once dried, glue edges , maybe test and see if a staple gun would go into the trim , if so then over and use staples to help secure your frame . Then attach to the mirror using command hook s Velcro hanging strips . Good luck !
Trim the baseboard to size and use liquid nail to adhear it to the mirror.