My psp 1000 had a similar problem (I think).
What I did was to re-solder the earphone jack to the sound board. The only connection that the jack has to the board are the soldered electrical leads, the ones furthest from the bottom of the psp had broken. If you can temporarily get your audio jack to work by lightlytapping or shaking the jack area then this is probably your problem.
1 remove battery
2 remove your memory card ( I don't know if you really need to do this but better safe than sorry)
3 remove the bezel
4 remove and unplug the home button strip
5 remove and unplug the display
6 remove the metal sub frame that the display snaps into
7 remove and unplug the button pads (left and right)
8 unplug and remove the speakers
9 remove the main board, the wifi antenna goes acros the top of the board and you will need to move it aside it is a close fit but it does fit ( A mining helmet and lamp may be appropriate now)
10 remove a small black plastic clip on the left side of the psp at the edge. (two screws)
11 remove the metal main board support frame
12 remove the sound jack board from the psp
13 soldering iron time.
14 examine the sound board, if the jack wiggles on the board the your fixing the correct problem
14 heat up the soldering iron. The idea is to get the iron really hot so that you can re-melt the solder on the board without buning any thing.
15 there are many small connections, touch each one in turn until the solder gets really shiny like a tiny blob of mercury. be careful not to leave the iron on the board too long (mystic blue smoke is never good)
the connections should be fixed now.
16 reverse the steps to re-assemble the psp
its just that easy!!!
I found this to be useful, it got me most of the way to the problem and helped me put it all back together PSP 1000 Speakers Replacement