Outlet in bathroom for lights....
Any pictures, model numbers, etc of the mirror/light unit that you could post? Firstly, the 2 prong cord, even if plugged into a GFCI, won't protect against shock if metal parts on the mirror unit become live. This mirror unit can't be meant for a bathroom or it wouldn't just have 2 prongs (likely wouldn't be a plug-in unit at all). The mirror unit would need to have all its metal pieces connected together and then conncted to ground, to be a properly grounded unit...which would then be connected with a 3 wires to a GFCI circuit, if used in a wet area. Tell us more.