You have carried out the basic tests already (including trying to use the computer with merely a spare mouse). Everything indicates that the problem is inside the LC II. Has the keyboard cable at some point possibly been disconnected while the computer was running? The ADB circuits can in certain cases become damaged if something like that happens. You may also want to inspect the logic board and its connectors/ports with a magnifying glass for obvious signs of failure (look for broken circuit board traces, bad solder joints and cracked/bulging/leaking/discoloured electrolytic capacitors and other electronic components).
A bad logic board battery should not have anything to do with this (unless leaking), but the 3.6 V voltage could be checked at the same time.