I ordered some of the 5 mil, shiny both sides as I couldn't find any matt, but it works really well, doesn't seem to slip, and even my hardest platens are improved. I never would have thought of this myself, thank you!

Are you winding the mainspring drum first?
Turn it with your hand a couple of turns anti-clockwise. The spring inside should start to tighten up and give you a little bit of space in that hole. Keep the drum wound up as you fit the knotted end of the drawstring. It might take you a couple of tries. If there is still no space in the hole, or only a tiny space, but the spring definitely has tension, you'll need to poke the knotted end in with something like a knitting needle.
(If you wind the drum and the spring can't keep tension then that's going to require a bigger fix/replacement, but hopefully this will be straightforward for you.)
Mind your fingers and good luck.

Ok, found it under grime on the frame underneath! 125267
Going by what I can see on the TDB it looks like it's a 1949 machine.

This is very like the project you propose. youtube video of typewriter with Comic Sans typeface fixed to existing typeface https://www.youtube.com/watch?v=R5a2hapPVMk

I read somewhere in the typosphere about using toothpaste and a toothbrush, so I gave it a go. It does take patience, but at least you know nothing horrible will come of it. Plus you get the minty aroma!