how to install S180-5 coil spring with 15-087 shoe

I'm sorry to hear you're having trouble. In my experience, you want to place the 15-087 into the opening first and hold it there while you install the 15-086 top housing. Once those are in, you just hook both springs to the 16-024 mounting bracket and then pull the whole assembly down. You might need needle nose pliers to do this.

That's how it's normally done, anyway. My associate Jody tells me you can't access the lower coil once they're installed. I'm wondering if maybe the cutout in the jamb doesn't allow you the room necessary. can you show me a picture of where this coil balance assembly is installing in your window? Can you also confirm that this is the right assembly? If you had something different, this hardware would not work to replace it.

I just looked at your older thread showing your springs. I don't see anything out of the ordinary. You want to first attach both springs to the bracket, then bring down the pivot shoe using a screwdriver. You want to bring it down 12", then turn the cam towards you to lock the shoe in place. That will let you reinstall the sash.