Replacing a mechnical switch with a digital one
It would be really helpful if you knew how the switch was wired within the KVM relative to the rest of the circuit to understand the best application of a transistor or IC.  

For instance, if the switch is just bringing an input to ground through a resistor, you may be able to sink the current directly through an I/O line

In the absence of specific information on what the switch is doing, you can't go wrong with replacing a switch with a relay.  The relay contacts would replace the switch, and you would drive the coil from the Arduino through a transistor (since the I/O line can't drive the relay directly).  Of course, you would have to deal with the clicking and the relatively shorter operational life of a relay over a transistor.