This happened to me once. Looked it up online and the easiest way to fix it I could find was to find something kind of small like a pen- maybe smaller, and put some glue on the end. Then stick it to whatever's stuck inside, wait for it to dry some so it will stick, and pull it out
This can, and probably will work.
The piece that is stuck wont effect the computer in any way, and your computer more than likely has ports in the front. If you have speakers, they will still work just fine regardless of the headphone jack being blocked. Buying a new soundcard would be silly.
What I want to know, is how that got stuck in there.