Please help with mounting a bathroom mirror

First... I'm VERY new to DIY and have limited skills, so forgive me if this is a stupid question or I fail to explain this properly.

I cut my cheap unframed mirror down to 25 x 20" and now I need to frame it and hang it.

I'm planning to use 1x3 poplar to frame it. I plan to rabbet the frame and hold the mirror in place with some brads or glue. Then I will cut the corners and use pocket holes to join them. I don't foresee any problems building the frame, my problem/ question is, 'how do I attach the mirror to the wall'?

I estimate the framed mirror will be approx. 10lbs, and I'm going to attach it to 1/2" drywall. I may be able to drill into a stud, but it is off centered, so it will only secure one side of the mirror.

I know I could use a wire and some D-rings or hooks, but I'm looking for a more secure solution since my wife will be constantly cleaning it.

I'm thinking I might be able to drill through the frame and into the drywall (maybe countersink the screws and use some type of caps to conceal the screws). What do you think??????

Any advice is GREATLY appreciated.... Thanks!!