Although I havent done the Sidewinder hack, I have done other hacks with 22 guage. I doubt the hole on the siewinder are smaller than those on a mouse (although it is certainly possible). I have found that twisting the bare ends tightly, then tinning them with solder, and then trimming the tinned ends to a short amount while cutting at an angle helps to get the wire through the hole.
Tinning keeps the individual strands from splaying out while you are trying to get the wire through the hole. Cutting at an angle gives you a smaller point to start the threading with.
FYI - tinning is the process of heating the bare end of the wire with the soldering iron and then applying solder so the wire (not the soldering tip). I use very fine silver solder for electronics from Radio Shack. I find that once heated enough the wire will wick up the solder, binding the strands together without adding to the thickness of the wire. So use as little solder as is necessary.
Hope this helps,
-Dave