"Any old dual KVM" - likely not.
Does the following link show the KVM:
https://www.amazon.com/IOGEAR-2-Port-Switch-Cables-GCS22U/dp/B001D1UTC4
Well/way backwards with respect to the connectivity that you most likely need.
$17 not likely to support your requirements. And, even if the switch happens to do so, the quality is likely to be suspect.
Switch is VGA. Your systems and monitors are likely DVI. Are your keyboard and mouse USB?
You want to keep the connectivity as simple as possible, free of extra adapters and/or converters.
That is why I suggested diagramming out the existing plugs and ports. Does not need to be a work of art. Just clear enough that you can figure out the necessary connections.
Then find a KVM switch that will match accordingly. Either directly or, if absolutely necessary, with some sort of adapter. And check if the switch comes with then necessary cables. Especially if the price seems low.
Here is a KVM switch that may be closer towards meeting your requirements:
https://www.bhphotovideo.com/c/product/1052046-REG/iogear_gcs922u_2_port_usb_dvi_cable.html
Representative only. Reviews not good (in my mind) and USB seems to be an issue. You should get USB 2.0 at a minimum. 3.0 if available and affordable. (Future proofing...)