Take a look:
Annette: I'm just getting ready to do the same! ...My wood is painted (both sides) and ready to go.
Quick and easy way it to attach wood to the mirror itself. Miter a 45 degree angle at the corners and attach with Loctite construction adhesive.
Here is a DIY frame bathroom mirror tutorial a friend of mine did on her blog: https://momhomeguide.com/diy-bathroom-mirror-one-room-challenge-week-4/
Just some ideas:
Here is the DIY for one I made on Homeroad https://www.homeroad.net/2018/04/framing-bathroom-mirror.html
If you didn’t get your desired answer, we’ll get some different ideas for you.
Have you thought of using "stick on moulding"?