I am replacing a mirror in a bathroom with a lighted mirror. When I removed the previous mirror there were 3 wires hanging there. 2 seem to be white and 1 black. The black is hot and when I put a meter on either white and the black I see 118V. The meter does not indicate a short between the 2 white wires (that I can tell).

Can I cap one of the white wires and just use the remaining white and black for the mirror? How would I ground it?

Was the old mirror hooked up to these wires were the two white wires together or separate ?

The three wires were loose and separate but were capped The old mirror did not have a light.

Also, I believe that the black lead is always hot. When I turn the overhead lamp off the meter still shows voltage.

Thaqnks for the help

I would turn off breakers off, one by one, until power goes of to the wires, temporarily hook a bulb to the wires may help identify, then with breaker identified, see what wire enter panel with that wire on that breaker. Should be able to see ground(green or bare) and neutral(white or gray). Many ways to identify from here.
2 dangers occur by disconnecting the neutral or ground . If their was a Ground Fault or leak to ground, disconnecting ground may become a live wire. Disconnecting the Neutral, on a multicircuit(2 hots on opposite phases sharing the same neutral)May cause a circuit to reach 240 volts. Both Dangerous, Avoid if you don't understand completely.
An Amprobe that clamps around a wire will not require disconnecting.
Place a load on two of the wires(hot and one of the whites) when the load is seen at the panel, see if gren or white, mark the same at other end.
You should end up with Black, white, and 1 taped with green. Good Luck, Take Care
Oh Yeah, If it is paint, try scraping.