I recently had a IIci with very similar symptoms. Pressing the power button would maybe give like half a second of life, or maybe do nothing at all. The problem was failed capacitors in the startup circuit on the logic board. Uniserver recapped the board, and now it starts up with no problems.

I've never tried it, but there's supposed to be a way to "jump start" a IIci whose startup circuit has failed this way, described here: https://mac68k.info/forums/thread.jspa?threadID=273You could give that a shot, to confirm whether the startup circuit is actually the problem, before investing time in recapping.

Edit: sorry, that link is for the PSU itself, not the startup circuit on the logic board. I can't seem to find a reference to the technique now, but I think it involved a jumper wire from the logic board battery to somewhere in that circuit. Maybe someone else will remember what I'm talking about.