before you try soldering try ordering a replacement connector from sparkfun...
Sparkfun's description of this part is a little confusing, but it is actually not intended to be a replacement for the ZIF connector on the PSP motherboard. Sparkfun sells these connectors to be used in other projects involving the Sharp LCD that is used in the PSP. According to VoX (who has worked with these), the Sparkfun ZIF connector has slightly different dimensions, and it is too wide to fit properly on the motherboard. The Hirose
FH19SC-40S-0.5SH(05) is a much better match, having ordered one of these connectors myself, I can verify that it is virtually identical to the one used in the PSP.
However, I would definitely try anthonyta's idea before you start ordering or replacing any parts, you can probably fix this problem very quickly and easily with the method he described.