I imagine that you used a solder pump to remove the old stick, judging by the surface damage here and there. But for this case that's a good thing: it means you didn't overheat the main chip or any other component near it. The only worrying damage I see is here in the picture:
Both analogs are related somehow to each other, and they get coordinated also somehow (I say somehow because I'm learning on the go; I don't know the specifics). If you have that signal missing, the other analog does crazy things. I also know that L2 and R2 are related to both analogs as well in some models.
Grind the line for that pin where I cropped the picture, then add a lot of solder trying to make sure that you recover the connection. If that works please come back and tell us. If it doesn't work, also come. We want to learn from this, too.
EDIT: Also, remember to press the reset button on the controller to rule that out when you're done soldering.
