Rewiring KVM Buttons
You can simply solder long wires to the back of the board where the buttons connect and put buttons on the ends of the wires.  If you get phantom switching (button presses when you weren't pressing a button) or several switches when you only press the button once then you'll have to look more carefully at the design of the circuit to eliminate false presses and properly debounce it.

But try long wires and buttons first - it should work fine.

The buttons you show have 4 pins. Connect your 2 wires to two diagonally opposite pins.