Window balance

Looks like you have a generic coil balance system, so knowing the brand probably won't help too much, anyway. You don't have to worry about that for now.

Your pivot shoe looks like a match for our 15-199, which itself is designed to be used with either the 1/2" thick Series 160 or 3/8" thick Series 165. I am confident that this hardware is what you need, you'll just have to determine what size coil is best for your window.