Wireless LED arrays are fairly new to the market but have decent kelvin temps and color rendering indexes. I just bought a fixture with an LED tube array and two LED spots on either side for over a bed headboard for reading. Runs on batteries. The tube in the middle and the two spots are on separate switches. Looks great. LED power draw is low but we shall see about battery life. The have similar fixtures for mirrors, framed art, etc. There are also things like sconces for mostly decorative applications. Just to give you an idea, this directional task lighting fixture runs on 3 "AAA" batteries.
Pulling wire for some hardwired vanity lights should not be a major problem. A track fixture of some kind might work but it looks like you have a small space. As mentioned, much will depend on how much is already on the that ceiling fixture circuit if that was what you plan to tap.