It's hard to tell from the pictures but it looks like the component failed and left its pads attached to the logic board. That's why it wasn't able to be soldered back on. Once a replacement is sourced, somebody skilled in electronics rework should be able to clean up the mounting pads on the logic board, and solder the new part on. @oldturkey03 's answer about sourcing parts should point you in the right direction.
I'm a bit worried that the attempt to reattach the old part damaged adjacent components on the board. Once you've sourced a replacement part, I would try to find a different local shop specializing in rework, and have them clean up the board / install the new part.