How can I repair a wobbly analog stick?

The problem with my GameCube Controller analog stick appears to come from the motherboard itself.

At default position, the analog stick will move with tilt, and between those points at which it will tilt (i.e. if I tilt it up and it goes to X, and if I tilt it up and it goes to Y, between X and Y), when I am moving the analog stick, it slides far too easily.

I have located the problem on the circuit board (the stick-like thing that connects to the analog stick isn't glued onto the two boundaries put in within the white ring inside the sensor), but I don't know how to fix it without heavy risk of either breaking the controller (spilling hot glue onto wire) or rendering the analog stick useless (gluing the stick-like thing to the white ring).

How do I fix this?