Have you tried different headphones plugged into it? Could it be that the jack is just dirty? Have you tried to clean the jack? Just asking.
That said, a reputable cell phone repair shop should be able to do this quickly and I would guess that the cost would be less than $100 if it is a cold solder joint and the jack is indeed in good shape. My son used to work for a shop and they routinely were fixing cold solder joints.