DC-in or logic board?

Hi! Sounds like it could be either. When you try to power on, does it chime, or does nothing happen at all, and the screen stays blank? If nothing, you might want to remove the topcase and verify that the power cable is in fact connected firmly to the motherboard socket. I've done a lot of troubleshooting on various machines only to discover that the topcase isn't even plugged in, or is loose. I'd even go so far as to try another topcase in order to rule out that the topcase itself is not defective.

The DC-in could be the culprit, although aside from the fact that the machine won't power up, it does sound like the DC-in is behaving correctly if the battery is bad. I would remove the battery, extended memory, and Airport, and see if you are able to reset the PMU at that point. If not, definitely try replacing the DC-in. To make that easier, just pop the bottom off the case, as well as the bottom shielding, and unplug the existing DC-in without removing it, plug in the replacement, and test it like that with the cable dangling and without bothering to put everything back together -- it will save you time an effort if in fact the DC-in is not the problem.

To test the battery more thoroughly, connect it to another iBook if you have one available, download the free CoconutBattery app, and see what that says. It will give you a definitive read on the battery capacity, current charge, and load cycles. However, make sure to leave the battery in the powered-on machine for about 10 minutes before trusting the readings in CoconutBattery, because sometimes it takes that long for a cold battery to "wake up" and start showing a correct reading.

If a replacement topcase produces the same result, you've removed all extraneous accessories, the PMU will not reset, and replacing the DC-in yields no positive result, then I'd say, congratulations, you've correctly diagnosed a bad motherboard! :-) But hopefully you'll find yourself with a working machine somewhere short of that. Good luck!