Don't these models have the little power switch on the back of the computer like the rest of the II series? It should be on the left side as you look at the back of the computer, a round pushbuttton with a large line down the middle.

I haven't played with a IIcx/ci for a while, but I'm pretty sure those things aren't battery-dependent. I could be wrong, though, so check the battery before you go tearing into anything. Next would be anything obviously wrong with the machine (liquid intrusions, missing/broken components, excessive dust, etc). Also, make sure the power supply is seated properly - sometimes they come loose or aren't inserted properly.