The Nylon rope broke

I think you're referring to the nylon string inside a channel balance. The string cannot be replaced by itself. You need the whole balance assembly. Take photos of the ends of the balance to show us, or use our channel balance guide to find the correct balance yourself.

Thanks for the photos! This is a Series 385 channel balance.

Your old 2640/"26 4" option was too strong! Choose 2630 and you'll have an easier time with the window. That explains why the string broke; too much tension.