It doesn't necessarily have to be broken. Maybe there's just some dirt in it that limits connectivity.
I don't know what you call these in English, but there are special cleansing chemicals for connectors and other contact surfaces. They remove any dirt and increase the connectivity.
Usually it works wonders. I once had a defective portable CD-Player (the audio sounded distorted and changing the volume produced loud, distorted noise). I fired "Kontakt 60" (probably the best stuff you can buy here in Germany for that purpose

) into the audio connector jack until it began to drop out on the other side of the casing, then put the device to dry for a day or so and it was working again.
However I think you should be more careful with "modern" Digital Audio Players, since they might have more fragile parts...