Check your KVM documentation.
Some keyboards also can be configured for legacy machines for better detection.
Check\change USB settings in the BIOS

Does it work in the BIOS or DOS?
Does it not work on 9x-2000? The last two KVM I bought had nothing but issues with this one so for these I connect directly.
Does it work on XP+?

I've had some success. With the mouse, anyway. I did things backwards, so I plugged the mouse into the retro first, and it installed loads of HID devices (5 or so) and it worked! Then back into the KVM switch, and it is working fine.

Keyboard is another story. It doesn't work....kind of. Tried the same method as the mouse, install it locally first, then to the KVM switch. But it didn't work, but I can use it in the BIOS and in DOS, and the RGB on the keyboard lights up. When going into Windows 98, it only works in safe mode (keyboard doesn't work there) or when booting normally it just hangs, has an exception and blue screens. It even now does it with the PS2 keyboard in alone....so it's going wrong somewhere.

So at the moment it works in BIOS + DOS, but crashes now all the time when using a new keyboard, and even the PS/2 one alone.

What motherboard is the retro PC using (Just to give us an idea of how good/buggy its Legacy USB support is)
I am assuming your using a USB cable between the PC and KVM for the keyboard and not a USB to PS2 converter on the PC end?

But something is definitely wrong with that PC if a plain PS2 keyboard plugged direct to the PC also doesn't work. That's low level stuff driver wise
Clean install of Windows 98 and no service packs will be the easiest base to start troubleshooting off