I don't have time right now to draw a diagram, but this should explain how the switch works. Assuming that you keep the original switch where it is on the power board, you can solder your new switch to the pins on the old switch according to this exquisitely rendered ascii diagram
.
(This is the PSP Power switch as seen from the back of the power board)
[``] <---Switch
[ ]
[-----------[__]-----------]
[ ]
[ ]
[--[ ]---[ ]------[ ]---[ ]--]
[ ] [ ] [ ] [ ]
[1] [2] [3] [4] <---Pins
Default Position: 2+3 Connection
Power On/Off/Standby: 2+4 Connection
Hold: 1+2 Connection