Why won't my iPod output audio?

Hello everyone. I have an old iPod 30 GB which I'm pretty sure is a 5th generation. About 5 years ago the iPod stopped outputting audio. This sudden loss in functionality did not follow any drops or water damage, since the device never left my car. I have tried reinstalling the firmware to no avail. As I'm typing this I am waiting for my computer to finish formatting the iPod so I can reinstall the firmware on a freshly formatted drive, but I doubt I am dealing with a software issue. For years I resigned myself to the belief that it was simply a bad headphone jack, but what got me questioning this theory is the fact that two years ago I bought a new Kia and got the iPod connector cable with it. Since this cable connects via the dock cable rather than the headphone jack I figured it would work. Sadly, no audio was outputted.

Could the problem be that the headphone jack is just bad, and is somehow effecting the audio output via the dock port?

Thank You