Cut about 3 inches off the end of the cable.
To make sure that it was indeed the connector, and not a cracked wire further up, remove the insulation of one set of wires. The device uses one set for each side. To remove the insulation, you can use a boxcutter, a sharp knife, a lighter, etc.
Attach the test lead from any MP3 player, as previously shown. If the reason for the non-functioning headphones was the connector, there should now be sound coming from the headphone speaker (driver). In this case, there was. Therefore, it was indeed the audio connector that was broken.