counter is white tile, and now the cabinet is white as well.
We believe the frame was likely put onto the mirror itself by the previous owners, using Liquid Nails. As you can see it is almost right up against the adjacent walls. Any ideas how to make this look presentable without having to spend $1000??? Would appreciate suggestions! 
I'm with them, paint it a satin or semi gloss white. It's a clean look.
Look into the mirror. Can you see the back of the trim reflected in it? If not, paint it white. If you can see the back, leave it. Painting it will not suddenly bring in lots of money...It is one mirror...
You could also paint it black like your towel. You can use a scraper to remove the paint that may get on the mirror.
https://www.justthewoods.com/paint-mirror-frame/
https://www.thesprucecrafts.com/paint-a-non-removable-mirror-frame-2879020
See if you can remove the frame from mirror altogether. May have to touch up the wall paint where the frame was. Mirror is probably glued to the wall.
Painting the frame of the mirror does not show what is behind to a great extent, as you have to get close to see. I would go ahead and mask off the mirror and paint it a beautiful white.
I know what they are talking about (seeing the original stain reflected in the mirror. My suggestion would be to add 1/4 round trim around the inside of the original trim (prepainted in the white...you can just glue this with white or clear liquid nails, and I believe this will prevent seeing the stained wood in the reflection.
Ignore it! The sale of your house will not hinge on a mirror! Make sure everything is super clean and decluttered and call it a day.
Perhaps you can paint the frame white, then use a white caulk where the frame meets the mirror so it doesn't reflect the small area on the back of the frame that can't be completely painted white because it is attached. Often buyers are not as picky as the Realtors, and plan to do their own changes and will overlook or even tear out some of the updates you've just done. Look at the big picture and don't drive yourself crazy over some of the details your Realtor is pointing out for changes. (I'm a former Realtor:) and we always give advice, but it doesn't always need to be followed LOL!)