I have a PS2 Slim whose power supply connector on the unit has come loose. Upon inspection of the board and adapter, the cause was obvious:
The solder on the middle prong had broke off from the board, but it also took the tiny metal ring that was set in the PCB itself with it, breaking it apart in pieces as well. Image of the board is below
I was given the advice elsewhere to just use a jump wire of approx. 12 AWG stranded to go from the main lead of the power jack to either the lead on the board (by lightly scratching off the solder mask to expose the copper beneath), or from the main lead directly to the resistor.
(For any of those that may have this problem in the future)
In soldering, cleanliness is critical. Just prior to soldering, clean your component leads. Clean your pads or lugs. Clean & tin your iron tip. Clean your solder too! Oxidation is not your friend. Beenthere's post describes good tip cleaning & tinning procedure. Isopropal alcohol and tissue paper works for cleaning everything but the iron. Try to find one of the "lint free" tissues.