Passing the spring cord through the washer will be initially easy, but when you've almost filled the washer with cord things become progressively more difficult. If you try to push the cord through a tight opening it's end will become blunt, especially if it's nylon, which is very soft.
There are a few ways to force the cord through tight spaces. For example, you can attach an improvised needle to the cord and push it through first, hauling the cord through behind it. Or, you could use a blunt metal pin to push the cord through by force. However, probably the best way is to taper the end of the spring cord and stiffen it with glue. This allows using the cord itself as a kind of a needle. This last method is the focus of this article. It is assumed that the spring cord is three-ply, but the same technique can be applied to any number of plies.
The process step-by-step Edit
Make two marks to the end of the cord sufficiently far away from each other. For the cheiroballistra you can set them 10cm apart; this is enough to get a good grip from below the washer, even if you initially manage to push only one ply through:
Unwind the plies up to the second mark:
Now cut the plies to taper the end. Leave only one ply between the end of the cord and the first mark, and two plies between the two marks. Beyond that the cord should be normal thickness, i.e. have all three plies:
Now spread out the ends of the plies:
Use a knife to taper the end of the plies:
Coat the end of the cord with lots of carpenters glue and twist it tight until the glue has dried slightly. If there are any steps in the cord after glue has dried, you can trim them with a knife.
Once the glue has completely hardened you can use the end of the cord as a needle.