How to continuity test without a multi-meter. The purpose of using a multimeter (in ohms mode) to continuity test is to associate pins and sockets with wire color so you know which ones to swap, and also to verify you've connected them properly.
You don't really need a multimeter. If you have a flashlight with a removable battery, for example, you can turn it into a continuity tester with some spare wire. And if you are modifying a PS/2 cable that's your source of spare wire.
You need three short pieces of wire, with insulation stripped off their ends. Using something as simple as Scotch tape, tape a separate wire to each end to the battery (don't connect the same wire to both ends, the wire will get hot and drain the battery.). Tape one of the wire ends from the battery to one connection on the flashlight bulb. Tape a third wire to the other connection on the lightbulb. You should now have two wire ends unconnected, one from the battery and one from the light bulb. If you touch those ends together the lightbulb should light up. Meaning you have proved continuity.
Now you can touch one wire end to a PS/2 pin and touch the (insulation stripped) cable wires and find out which color wire lights up the bulb. You have now mapped pins to wire color.