Step 1: Diagnose the Cause of the Failure
Turn it on. If the motor turns, but the brush does not oscillate, this means the gearbox has failed, but everything else is OK. If the motor does not turn it could be either the motor or battery that has failed. In this case, you will have to disassemble the unit either by cutting the housing or carefully unscrewing the base from the brush housing. The picture shows how the housing was cut. Note that I am saving the switch assembly.
Carefully cut the outer shell of the housing with a saw and carefully remove the components. First check the battery voltage. If it is less than a volt after being charged, you will have to replace the battery. If the battery shows more than 1.5VDC then the charger and battery are OK. At this point, you should test whatever LED you plan to use and see if it lights up at your battery voltage. Don't forget to use a current limiting resistor. You can check the motor, but it will not be used on this project.
Now is a good time to check the on-off switch to see if it can be used or if you will have to add your own switch as shown in photo 2.
(This is a different unit that I modified previously and shows the mini toggle switch that I added)
Step 2: Adding the Extension Wires
On the base or battery end of the heavy wire, you should solder some light ( 22-26) gauge stranded wire to the end of the heavy wire then run the light gauge wire to the battery and switch before securing the heavy gage wire to the housing. After preparing the ends of the wire, secure them to the housing as shown in photo 2. I used epoxy, but you could use hot glue or nuts and bolts.
Step 3: Wiring the Complete Circuit
If you had cut the top off of the unit, you should insert the heavy gauge wires through the shaft opening before soldering the LED. You should now test the circuit before attaching the top or inserting into the housing. I used epoxy to re-attach the top.
Step 4: Ta Da - the Results of Your Work!
Good luck on building your very own, recycled "TOOTH TORCH" !