Jack snapped in two

It looks like the jack is a two-pole stereo jack. This fix will require some soldering.

1. Find a 3.5mm (AUX) cable of sufficient thickness.
2. Cut the end off of the headset's cord, and cut off about 3 inches from the end of the AUX cable.
3. Strip about an inch off of the outside sheath of both cords, then strip about 3/8 inch off of the end of all the ends of all wires inside (there should be 2, plus a sheath or ground), and twist each internal wire so that it doesn't fray. Twist the metal sheath into a third wire.
4. Slip on three 1.5" pieces of appropriately sized heat shrink tubing onto each wire, and slip on a fourth 2.5" piece over the entire thing.
5. Match the colors of the wire to the end of the AUX cord, then solder the two together.
6. Slip the heat shrink tubing that you previously put on over and use a hair dryer, heat gun or lighter to shrink it around.