No offense intended to tjohnusa, but you don't need a soldering iron for this problem and you can totally do it yourself.
Depending on where you are, if you check craigslist chances are you'll find someone willing to repair it for you for $30 - $50. You can go the DIY route and fix it yourself like some others mentioned above (not the tin foil one though).
At first I thought this was the same problem I had and was going to offer to fix it for you, but my problem is, I guess, rarer where the headphone out wasn't working because the solder joint on some part on the main board was bad.
If you can get sound just fine by pushing the headphone connector, as you say it's probably just the headphone jack needing to be replaced. It's super easy, you just have to be careful with what you do, and be extra careful when you open it. The 4G has a single cable attached to the back, but the 5G has 2 cables on opposite sides, so you have to carefully open it a little and remove one cable before opening it completely. It's not hard at all, just a little daunting if you've never done things like these before. But the worst thing that can happen when you're opening up the unit are
1. You break one of the cables, ~$10 for a replacement cable.
2. You scratch the heck out of the front plastic, ~$10 for replacement plastic if it's deep gouges, otherwise you can get a plastic polish for $10 and smooth it out (and have it around for other things you scratch)
3. You scratch the heck out of the front plastic (deep gouges) and you bend the heck out of the rear panel, ~$20 for replacement front plastic and rear panel.
It's really easy to do yourself, and the part you need is about $8 - $10 off ebay shipped. I have one because I initially thought I had the same problem as you but it turned out not to be the case.