You state that it only works from a charged battery...by that do you mean that the AC adapter does not power the computer when the battery is removed? What color is the light on the AC in any given situation? If the AC adapter powers the computer when the battery is out, and you are certain that the DC-in is fine (which it usually is -- they don't often go bad in A1181s), I'd try replacing the battery connector. Make sure to get the correct battery connector -- some are Energy Star, and some are non-Energy Star.
If no AC adapter will power the computer when the battery is out, I'd verify the DC-in is good by swapping with a known-good DC-in. Beyond that, it's probably the logic board.
Also, stupid question, but have you verified in another machine that the battery you're using does in fact charge? Many of these battery issues end up being...the battery. Yours sounds a little more complicated though.
I don't know how to "inject" power to the logic board, and I'd be very reluctant to try that, for fear of frying the board.