Step 1: Take Apart the Light
The board with the LEDs, switch, and resistor is easy to pop off; just make sure it will be able to sit back in place.
You may need to break of some unnecessary corner of the board to allow space for the wire that you will be running from the bottom of the board.
Step 2: Mod the Switch
Reassemble the switch. Make sure the spring and the plunger go back in the proper places. Test that the switch is normally off (no conductivity) but on when pressed (conductivity). Also test that the light goes on when the batteries are in and the switch is pressed.
Step 3: Solder on the External Connector
Figure out which side of the switch controls the light, and solder wires to the matching contact points on the other side of the switch. You can use wires from inside Cat5 (ethernet) cables, or any two (or more) conductor wire.
You can use short wires connected to a jack that you'll mount on the case, or longer wires to a plug (as show in the photo).
Drill a hole through the case for the wires (or the jack). Make sure there's enough room for everything to fit back together. Also make sure you pass the wires through the hole before both ends are soldered.
Step 4: Reassemble Everything
At work we use these for students with limited dexterity, they are connected to mice so the button becomes the left click.