I just picked up a USED (no warranty) Mid 2010 MBP with this issue. I have read the many threads all over the web and tried EVERYTHING and nothing has worked. Even headphones are flaky, you have to jiggle them to get audio so it sure seems like a bad jack. I used to a great solderer but at my "advanced age" I'm not as steady as I can be.
I've tried (among other things):
Blowing into the jack with a can of strong compressed air
Diddling with toothpick, q-tip (cotton removed) and any other device I can insert into the hole. 🙂
Twisting and shouting, shaking and cajoling. No luck.
I have a fresh clean install of Mavericks on a new SSD with nothing installed so it isn't a software problem.
I wish I knew which soldering pads on the mother board need to be bypassed or shorted to disable the switch as I have no use for the headphone jack. I primarily want to be able to use the speakers for Skype.
Seeing as I paid under $300 for the MBP it doesn't make sense to have Apple repair it as it would probably cost twice that, Other then the jack the MBP is excellent and was a good deal.
What I don't understand is why the speakers work for the startup sounds.