I don't get it. Did the port itself come off (like the hole moves around, or is crooked) or did the plug end of a pair of headphones get stuck and then snap off inside? if it's the former, yeah, you can try resoldering them, but be careful. It's part of the logic board, which is the board that contains everything in your system-- it's like it's central nervous system.
If it's the latter, it could probably be extracted. Sometimes the jacks (female end) have an opening at the other end, you could push something in to get it out (obviously you'd have to take your computer apart first)...
you might consider finding a local computer repair service, see if they can do one of these things for you cheaper than $1000-- just make sure they're bonded for work they do, you don't wanna lose the whole system!
pnw