Hi Karen Lynn,
We sailed in a Sky Suite 2 weeks ago on the Eclipse. While the hair dryer was in a drawer in the bathroom, there was no lighted mirror. There were the 3 mirrors in the bath-the one straight on and the other 2 on an angle fronting each storage area door. There is also a light over the mirror at the desk area. I'm sure if you ask, your butler will know if there is one in the room or he/she can bring you one.
Also, if you're headed to Alaska, if your butler doesn't bring them, request the plaid blankets. They're great for sitting on the veranda!