How do put led lights on a dresser?

If you're thinking of "string lights", then I would look at different kinds of trim to cut the glare into my eyes and to hid the wiring.

How thick is your mirror? Is it wood construction for attaching the lights to? Are you OK with tacking cable staples, brackets, nails, etc to your mirror?

The trim could range from "L" shaped wood (pricey), to "L" shaped metal flashing (which requires cutting with tin snips or tin shears and a steady hand),

to something like 4" PVC plumbing pipe (requires a table saw, IMO) and cut a 1" channel in it and glue up the corners with 90 degree angles.

The PVC plumbing pipe idea could be made free standing if you don't want to tack anything on to your mirror and I can sketch what I'd do in that case. You'd be best off if you could borrow a PVC pipe cutter or a hack saw, the primer and the glue, but they're not terribly expensive and can be used to make a lot of cool things.

You'll need some small cable staples and if you don't have an in-line switch for the string lights, then I would look at either a small switched surge protector or buy an in-line switch and cut the string lights and wire it in. If you have a switched outlet in the room, you're good to go.

Hi. Did you know there are attractive battery operated sconces that you can attach to your mirror to bring all the light you need? Use foam tape to attach. The sconces can be found online at very reasonable prices. Good luck!

LED tape lights can be put under the narrowest lip and concealed

I would tape them on or hot glue or get Christmas light clips for gutters they should work to clip on