Soldering AC wires to this PCB
I doubt that cable is carrying household voltage. It's probably DC voltage in the 12-24 volt range from a power supply brick.

Use your multimeter to figure out which part of the connector on the cable is positive and negative. Then you can desolder the socket from the board, cut the connector off the cord, use the multimeter to identify positive/negative in the cable and solder it to the circuit board.

If I had to guess, the two pins sharing a trace are probably positive, and the third pin looks to be attached to the ground plane and therefore probably negative.