How can I clean out an audio jack?

Haha, been there done that... Chili not soup though.

First, let me say that you may have damage to the headphone jack circuit circuit that's actually detecting the headphones. I can't find any teardown info on your phone and have never torn one down, but usually the jack is attached to the motherboard via a flexible circuit that "senses" if the headphones are plugged in. Liquid spill on it will cause rust, which causes the circuit to act up. If this is the case here you'll need to disassemble the phone, remove the headphone flex, and clean it. If the components haven't completely shorted and been damaged (which usually doesn't happen) it'll be ok. Otherwise you'll need to replace it.

To clean without disassembling buy some Electrical Contact Cleaner. Pretty much any brand should be OK. It's basically a spray can with Isopropyl alcohol and sometimes other solvents like MEK, etc. Anyway, when you spray the stuff it should be clear liquid that evaporates very quickly and leaves no residue. It should also have one of those red plastic tubes that let you spray it in a small hole (like your headphone jack!). Test it on something before using it on your phone.

To use it on your phone orient the phone so the headphone jack is facing down. Use the plastic tube on the spray can. Put the tube up into the headphone jack and spray for about a second then stop. Let it drain out. Repeat. Do it several times and it should drain everything that was up there. You can also intersperse this method with using a cotton swab with some of the cotton removed from the tip (so that it'll fit in the hole) to push straight up and down in the hole to clean it out. Don't rotate the swab as it can grab the ring or sleeve contacts and bend them.