It looks like the jack is a two-pole stereo jack. This fix will require some soldering.
# Find a 3.5mm (AUX) cable of sufficient thickness.
# Cut the end off of the headset's cord, and cut off about 3 inches from the end of the AUX cable.
# 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.
# 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.
# Match the colors of the wire to the end of the AUX cord, then solder the two together.
# Slip the heat shrink tubing that you previously put on over and use a hair dryer, heat gun or lighter to shrink it around.