Luke, hello:
Looking at your first photo, that solder isn't that bad or hard to fix. Buy a 'braid' which comes with solder irons. To pull off extra solder, you lay the braid end down onto the joint, and then push the (hot) iron tip into the braid surface, with a bit of pressure.
Wait for a few seconds, and that wire braid will pull the solder up, into itself. That means that you've transferred the solder, while it's melted. Might be a good idea to clean that area at that point.
For the braid, simply clip off the little end that has solder in it now...discard.
You need to then scrutinize the traces...not that hard. You can observe how the traces don't touch, so logically any 'solder bridge' will be a problem.
Might be time to start learning some basics. Can you buy a multimeter ?
Using ohms scale, you can at least check for shorted traces. But keep in mind, that if that jack has a switch contact, the two lines from the switch will appear as 'shorted' or connected. In fact, if you are curious, could try inserting a plug in the jack, as that should push any switch to an open position.
I don't know, but that plug inserted SHOULD cause (any) switch to show as 'infinite' or open circuit, a correct result.
It's troubling that a wiggle, of the inserted plug, would cause flakey contact...that might indicate Jack is bad / worn. If you can figure out the two contacts needed, you could, optionally, arrange to mount a fresh new JACK, put that in a secure mount, or make a secure mounting nearby, for your use.
That is ground, and signal that you would be using, to create an alternate. You might be able to test your new mounted jack, temporarily, and then solder the wires better.