Using the 3640 in the 3000 desktop was never an officially endorsed solution by Commodore due to heat issues - it's not too safe to run without a heatsink, and units with a good one have trouble fitting physically. I'm assuming you have one with a tiny heatsink with integrated fan? Hopefully things aren't getting too hot.
Anyway, I'm afraid that disassembly is going to be pretty likely. The 3000 has what seems like a gazillion jumpers that need to be set for CPU cards. Make sure those are all correct. Also, what are your Ramsey and DMAC versions? 7 and 4, respectively are best for a 3640. I couldn't get one working in a 3000T with Ramsey 4 and DMAC 2.
But there's a slight chance you don't need to take everything apart. It sounds like your keyboard is plugged into a KVM and a PS2->Amiga adaptor. Try without the KVM. Get as direct a connection as possible to eliminate other possible sources of trouble.
