I agree, check the plug and verify it's connecting on all pins.
Next, since the power button seems to be working, I would try resetting the PMU. With the power off and while connected to AC, hold the power button in for about 10 seconds, until you hear a tone and see the sleep light flashing. I'm wondering if this might cause it to start working.
There is only one cable from the topcase to the board, so the odd thing is that the power button works, but that the keyboard doesn't. Is this a new topcase, or a used topcase? Have you witnessed this topcase working in another laptop?
I have seen Software Update doing a "keyboard firmware" update when I have attached an older (1.0?) topcase to newer A1181, I assume to update it to the newer (2.0?) topcase firmware. While I'm not convinced that the firmware of the topcase is the issue, since I've never had a problem with any topcase working on any A1181, I would be curious to if by connecting a USB mouse/keyboard to the computer and running Software Update, you might be able to update the topcase firmware, and see what that does.
Again, it's a stretch, but besides the keyboard just being bad, that's all that comes to mind.