Flip the board over.

Locate the joystick solder nubs on the back of the bottom of the board.

Using the soldering iron, heat it up to around 350-375 degrees Fahrenheit and lay the tip on the aforementioned nubs.

Hold it for around 5-7 seconds or until the solder nub liquefies. Repeat this for the rest of the nubs.

Having done this, the joystick should slide out the board. Install the replacement joystick by sliding in the joystick prongs into the slots that the old joystick sat in.

With the joystick in, solder the prongs until they are secured into place on the boards.