Grab a bin to store all the headphone parts. Even a big zip-top bag works! Think ahead about how you're going to keep all the parts together, especially if you're not doing the project all in one day.
Remove the foam pads from the ears and store them for later. Use a screwdriver to remove the earphones, being careful not to rip the delicate wires soldered to them (you can always re-solder it if one does break)
Drill a hole in the top of the earphones big enough for a three-conductor piece of ribbon wire to go through. This will connect the output of one LED ring to the input of the other. Leave plenty of slack in the wire, as you can always store it in the earphone but can't add more length later.