It probably won't be the battery causing the problems -- both of my IIcis boot fine without a battery installed. It's true that some of the other II series Macs won't boot with a battery, but the IIci should be fine without one. It is kind of weird that your battery is already down to 2.5V and it's brand new, though -- very odd.

I had a similar problem with my IIci that hasn't been recapped yet -- it was gunk on the motherboard near the power circuit causing it. After I blew some compressed air on the motherboard near the power circuit it started working again. Since you just recapped and washed it, it's probably not gunk -- but it is probably something in that area of the board. I would double-check for anything weird (gunk, corrosion, loose solder joints) on the area of the motherboard below where the power supply goes -- that's where the power-on circuit is.

When I was having that same problem, Dennis Nedry pointed out if I had fried the CPU for example, the power supply fan should still come on. So since your motherboard is doing *nothing* at all, it's very likely that the problem is in the power circuit, and less likely that you have zapped any ICs or anything like that.