If it were me I would remove the caster insert from the legs and fill the holes with dowels. Then because a bath floor gets wet put nylon tacks on the bottom of the legs to elevate the cabinet a quarter of an inch or so from the floor. Can't tell from the picture, if it's red oak getting it wet causes it to turn black.
Since the height is a little low you might consider extending the length of the legs and put a piece of trim around the seam and at the bottom. Once done it should look like it was always like that. The trend today is to make vanities 36" tall so if you have enough height with the mirror you might try to get closer to that.
The mirror, usually those the glass is held in by corner blocks with a back to cover the opening. If you're mirror is anything like that you might just leave the back off so it will fit flush with the wall. The frame could also have the mounting brackets removed and attached to the wall. You could use something like keyhole hangers to mount the components. That way if you needed to get to the glass or remove everything to paint it should make it easy.