No identification can be found on any hardware,

Hello! Looks like you have an old Crossly window balance. From your photo, the length of the metal channel is 23", so check out our S700-23.

I recommend replacing the balance entirely, rather than trying to find the string by itself. You likely won't find the right type but if you did, we don't recommend trying to repair it. These are delicate systems and aside from the spring tension never quite being right, there is a risk of injury. I've known people who have tried to repair a balance who have had the spring jump back and hit them. Simply replacing the balance is less time consuming and dangerous.