I actually had to hardwire my friends analog stick because his pads werent doing the job. Ill try to explain how to.
Under the metal framing of the psp there should be 4 gold square plates on the motherboard, if you were to place the analog nub on it like how it would sit in the case, that is how you would want to connect the contact plates, use some short wires to connect the 4 plates on the motherboard and the analog nub. If you still arent sure i can post an image tomorroe