The problem here is, that most ps2->amiga converters (especially keyboard) don't work properly with intelligent KVM switches.. I guess the KVMs don't handle some command that the converter wants to send to the "keyboard" and then the converter thinks the keyboard disappeared..
If you find an ancient wires only KVM, that will work, as it's only connecting different ports together.
These adapters are claimed to work with KVMs, but you'll have to take their word for it.
http://www.mrhardwarecomputers.com/pages/mousekey.htm