If other headphones function perfectly I don't see how it could be a contract problem with the jack, unless the headphones are visibly damaged or even bent at that end.
As for isolating the fault, the simplest, most certain thing to do seems to be to replace the length of the wire. I don't think iPod earphones upon and close without a lot of visible damage, so if it is something inside the speaker the even replacing it might make the repaired earphone uncomfortable in your ear.