Why, of course! You're looking for a "magnetic catch".
Just you should possibly install it the other way around - a metal plate on the wall, and the catch screwed on the mirror.
Images added:
- Magnet mounted as plug in the cabinet with large head screw contact
- Magnet in cup washer and mating contact washer mounted with screws
- Thin magnet to metal plate
So, I went with the mastic.
Bad news - between not boing thick enough, and the fact it's pressboard, I didn't feel confident using any kind of screws to hold the door. I went with a pure mastic solutions.
Good news - this in fact a door, so you can take it off. This lets you keep it horizontal, and put weight on it to help the glue hold.
So, I took the door off the hinges and laid it flat. I marked off the perimeter of where I wanted the mirror with blue tape (Blue Painters tape is one of my favorite tools). I thought the surface was a bit plasticky still, so I scuffed it up some with some sandpaper. I put the mastic on the mirror, trying to keep the inch from the edge. By the time I was done with my pattern, I was almost done with the mastic tube, so I just emptied it. I put some 2x4s and other heavy things on top to help the glue hold (another advantage of having a mirror it on a door). I let it sit for the recommended 48 hr set period and put the mirror on.
The only downside I see so far, is the weight on the door. It seems stable, but the wardrobe didn't seem 100% level. What was OK before, what closed fine before, now stays open a bit with the extra weight. My next subproject is re-leveling the wardrobe.
Best Answer
You haven't given us much to go on from those pictures, but having assembled and installed a headboard on my bead recently with a similar type system:
Those rails do not look sturdy enough to support the entire weight of a mirror and prevent it from falling over. I'd submit that the mirror should be resting on the dresser below and that the rails are simply to keep it in place.
It does appear that the screws through the brackets are intended to go into the holes shown in the picture of the rails, but I'd suggest that those holes are actually to mount the rails to the back of the dresser and that the screws in the brackets simply hold the rails in position once the mirror has been set on the dresser.
The top of the rail should mount near the top bracket on the mirror. Then they should extend as far as possible below the edge of the mirror that they can provide solid leverage where screwed into the dresser to act as a lever to keep the mirror from falling over.