I wonder if there's not a "wireless remote control" solution to this that will not involve any drilling or cutting. Something like the remotes for 12vdc vent fans. Wire the remote in parrallel to the existing switch so either will turn on the light. Then the remote could be kept in the kids bedroom and they could turn on the light from their bed so they can see how to get to the bathroom.
Of course, being kids, they could also loose the remote.
Another thought: install a string-pull switch in the end of the fixture. The string will hang down enough for the kids to reach it. You could replace the original with a string-pull if having the string in front of the mirror wouldn't drive you nuts. (It would me!!)