Once all the contacts are desoldered, the old joystick can be removed. This may take a bit of practice since all the solder will have to be removed. It does help to pull a bit on the joystick while melting the solder and using the wick.
Ensure that the contacts of the replacement joystick are identical to the original.
Check that all the holes are cleared of old solder. Hypodermic needles as well as very small drill bits can be used to clear the holes. The molten looking substance on the circuit board is the flux used for the desoldering.