Same situation with my iPhone 5. Installed a new battery 4 months ago and it has been fine until now. I read a post on the apple forums that suggested toggling the silent switch a few times. I have absolutely no idea why this sort of sorcery worked, but it totally fixed the problem.
I recently repaired my PowerBook G4 15" Aluminum 1.0GHz DC/Usb Soundcard for the exact same reason. It is NOT possible to push the stem out from behind the jack because it is enclosed. I removed the stem by lying on the ground, computer partly supported by my coffee table with the dc part off the edge, and used a safety pin to depress the latch that is inside the jack, then wiggled the computer around until the stem came into center view. I used very sharp tipped tweezers to grasp the stem and then pulled it out. The tweezers are NOT house-hold type tweezers but the type used to work on electronics. Hopefully your board is not damaged. Hope this helps. Update: Did you try a replacement charger to see if it works? Sadly, I had to replace my USB/DC Soundboard in the the end