Make sure that you check with another headphone, just to make sure that the ones you have work. You can also take a can of Duster and hold the iPod upside down and spray a few puffs in the head phone jack. If it still does not work, I think it is time to open up the iPod and replace the head phone jack. You can find the guides and the parts on here. Good Luck.
Includes headphone jack & hold switch — Currently out of stock.