This procedure is pretty simple and only takes a couple of minutes- but it does require steady hands!
Step 1: Tools
1) A Dremel tool- the smaller rechargeable variable speed model is best but any model should work
2) A 1.0mm hart burr -available here: http://www.cooltools.us/45-Degree-Hart-Bur-1mm-p/drl-324.htm
3) Lots of light (good magnifiers don't hurt either!)
Step 2: Cut a Notch and Remove the Broken Part
Turn off your Dremel tool and remove the burr. Turn the phone upside down to remove any metal particles. Now place the burr back into the jack socket and gently feel around until the burr is located in the notch you cut. Apply a little side pressure and carefully pull the burr out and the broken jack part should pop right out with it.
I have yet to see a headphone jack that is broken off lower than the one pictured here so hopefully this works as well for other people as it has for me.
Please let me know if there are any questions- thanks!