I'm definitely not going to get mad, rather I'll take the opportunity to say hi.
Regarding the topic of this thread I don't know how to help this user: I am an expert of psp software more than of its hardware.
Hmm, have you tried to charge a battery full without the psp used?
Just i'm not in anyway specialised in hw related to psp, i only once modded one with a homebrew pandora battery, but here's a few thoughts as a mechanic:
Have you tried modding latest way with pandora battery? (for it might be a f*cked firmware).
Have you tried updating the psp normall firmware? (for it might be a f*cked firmware).
Have you got a replacement battery circuit board to test? (this for this board is responsible for either charging the battery or switching to ac power (cutting the battery power off).
Have you got a volt-meter or multimeter checking the battery power and ac power are about the same voltage? (the AC plug should give a little more than the battery).
If you got a watt meter (or can calculate those by working math through the amps and voltage of both the circuit of the AC to give more Watt's than the battery can give fully charged. (if not your AC is under powered, might mean the diode bridge is broken or there's a leaky cap).
solder splash should be removed if it's out of place (please make a picture of this if you can before removing it). each soldered place shouldbe cleaned with a brush and alcohol, but watch out for any plastic part or the LCD, for they might react to alcohol in a bad way.
Check all transistors/diodes to function like a diode, and if they have at least 10 Ohm resistence.
When checking fix-it.com it's almost always a dying battery it seems.