There's probably a small hardware switch in the headphone port (like in many Apple devices) that's stuck. When there isn't anything plugged into the port the switch should be protruding ever so slightly. If you have a flashlight handy you can peek inside it and see if you see it. If not, try plugging in a pair of headphones a couple of times to see if it unjams it. If that doesn't work, it could be a software problem or a deeper hardware problem.