My headphone broke into my headphone jack...

I would recommend against trying to do this yourself. You' could just make it worse.

My experience has been that, even if your device is out of warranty, if it's something that can be fixed relatively quickly without new parts, the folks at the Genius Bar will do what they can to help. I had a MacBook Air about 5 years out of warranty when the touchpad stopped working properly. I took it in for evaluation. The person who helped me said that it could well be just a set screw. He took it into the back, returned in about 10 minutes and it was working perfectly. It had been just that small adjustment. They didn't charge me.

Make a Genius Bar appointment at your nearest store and have them look at it. There is no set fee for this and they may or may not be able to help you remove it.

If you need to replace the port, I'm guessing about $30. But this happened to me before on an iPad. Try using tweezers to pull the cord out of the jack.