The Logans wrote:
You're right, there is no way to pull power through the walls or ceiling, because of all of the insulation.
Actually, running wiring in the ceilings and walls (except walls with solid foam insulation) is fairly easy. I've done a LOT of it now.
Insulation is not placed so that it fully fills a ceiling cavity. The factory runs all the wiring loose in the ceilings and it just lays on top of trusses under the blanket of insulation. Often, the insulation is poorly done anyway and there's lot of voids. You may not even find any near exterior walls. It's surprisingly simple to add wiring once you've tried doing some - romex, coax or LV wiring.
For lighting at the mirror, I suggest installing one or two "puck" style LED lights in the ceiling so you won't have to get wiring into a wall. Cheap on ebay. Check lumen output first tho. Our TT has two factory installed puck lights over our mirror and they do a good job of lighting. Could be a little brighter tho. I added a puck light in our shower (it was too dark) and maybe you could do the same while adding lighting over the mirror.
To run wiring in the ceiling, pull down lights, AC grilles, skylight trim, speakers, etc. to give access into the ceiling cavity. Use coat hanger wire or a fish tape (cheap at HF) to leapfrog the wire from access point to access point. If needed, you can enlarge the hole behind a surface mounted light to give easier access as the light will cover the hole. I've fished as far as about 6' through the ceiling in one go. You probably won't need to, but wiring can also be run inside cabinetry, under showers & tubs, inside fridge enclosures, etc.
If you have something mounted on a wall like a switch, receptacle or holding tank monitor panel, there is most likely an existing hole in the top plate right above that the factory used to get wires from the wall into the ceiling. If installing a small wire/cable and not romex, you can probably run the wiring through the existing hole no problem.
Good luck and don't give in too easily!