coil spring system replacement

This looks like a 15-140 Pivot Shoe, which uses Series 150 Coil Balances. For the housing hardware, see our 15-143, 15-144, and 15-145. 

If one of the coil balances is stamped 5, then it's like an S150-5 5 lb spring. If you're uncertain, give us the exact weight of the window sash, and we'll work out what you need. 

To install these coil springs and pivot shoe, You would use the 15-143 or 15-144 housing and 15-145 bushing (if you can't reuse your originals) and hook the spring to the 15-140 pivot shoe. The loop at the end of the spring should just clip into the s lot in the pivot shoe. For two springs, you would place one above the other and work them into the cutout in the window. The installation can be a little tricky, since the originals where installed by dropping them down the channel before putting the window frame together, but it is possible. It just takes some patience and maneuvering.

Sorry to hear you're having trouble. I am not quite sure what you mean by replacing the shoe, as the shoe was A) already removed in your original photos, and B) it does not go into the window sash, but rather the jamb track along the sides of the window frame. Did you mean you're trying to replace the pivot bar? If so, the first video below shows how. The second shows how to remove and replace Coil Balance systems.