There are two wires going into the boom. One powers the LED lights and the other goes to the microphone. Each wire has two smaller ones in them.
Cut both wires, strip the shielding, solder the new wires together, and wrap them in electrical tape or use heat shrink. Be sure to insulate the wires from each other.
The two sets of wires are different colors so you do not need to worry about figuring out which wire powers the LEDs and which goes to the microphone.