For a good Dressing Room / Make-up Room experience your solution should consider the surface colors for floor, ceiling, walls, millwork as they all affect the reflected light that arrives at the performer as they apply make-up and/or try to select fabric colors. The color content of ALL the lights in the room similarly affect the outcome, so refurbishment of the ceiling lights and mirror lights must go hand-in-hand.
The lights should be high color rendering index (CRI) and a consistent color temperature (CCT) that is similar to your incandescent stage lights (even if you are transitioning to LED stage lights).
Other considerations are that there are special requirements in the NEC (NFPA 70 - National Electric Code) for Dressing Room / Make-up Room light fixtures, their switching, and the monitoring their operational status. Getting all of these factors right is easier said than done, as most electricians and purchasing agents are not aware of these special requirements.
The mirrors, lights, surface colors, and millwork are a system, not just a bunch of parts thrown together.
Feel free to PM me if you are interested in a consult.