I have had this problem myself and found it fairly easy to fix so here's the Problem and Solution:
Problem:
When connected to external power source, the orange 'charging' light doesnt turn on, and under 'battery information' the charge level shows as 0%, but (even plugged into the power source) the battery status says Charge complete, hours left says -, and power source says external, when the external power source is disconnected, the psp shuts off instantly.
This i can only assume is an error with the charge detection system of the psp, which seems to happen when the psp battery level is at 0%
Solution :
Remove the battery from the psp unit, and taking two sections of wire, connect them to the slots (+) and ( - ) of the battery pack, and either ;
a) Connect the other ends together for a short ammount of time (shorting it), or
b) place the other ends on your tongue for about 3-5 seconds (its a low voltage so it wont hurt, just a small tingle)
If you do not wish to put your tonge on to the wires and the words "shorting it" scare you then you can touch the wired against some Tin Foil (any brand, theres no difference)
Once done, place the battery in the psp, and connect the external power source, a second or so later the orange charge light should come on and everything will be well.
Play:
you can now get back to playing your games and doing what you do with your life.