Need ideas on how to rig this up

I would probably remove the further away light and replace it with an electric motion detector made for outdoors. They take 2 bulbs and you can adjust how long they will stay on after the movement stops. They will be on as long as anyone is in the room and moving and can be set to stay on for 30 minutes after all movement stops.

If that is not enough light, your mother can pull the cord on the other light. There is no reason to use a battery powered light since you have electricity wired in already.

I have one light that is on a wireless switch. It is for a ceiling fan and has buttons for 4 fan speeds and light off or on. So, that is possible, too. It does involve some minor wiring work both at the light and at the switch. So you wouldn't have to have a switch wired in in order to have a switch.

A wired switch wouldn't be all that hard as long as you are willing to have the wiring in a channel mounted on the surface of the wall and don't have to have it installed inside the wall. That should not be too bad a problem in the laundry room. It doesn't look all that bad and you generally don't entertain guests in teh laundry room, anyway.

put a eye screw in the wall by the door, and extend the pull sting to the eye screw and let it hang down the wall by the door, (have done this in my barn on a few places,) you may have to remove the porcelain holder and rotate it to the sting is more in line with the door,