Coil Balance or Spiral Balance Replacement

Hello! This is definitely a weird setup. Your first picture shoes a coil balance, but your second picture shows a pivot shoe specifically made for spiral balances. These should not be compatible. I don't even think they're the same size; your pivot shoe is similar in style to our 15-045, which is very skinny at 3/8" thick. Meanwhile your coil spring looks much thicker, maybe even approaching 1/2".

How do these function together? Is the pivot shoe loose in the frame? How does the spring connect to it? It would be a huge help if you could give us as much information as possible on how these function. Spare no detail.

In the meantime, I can tell you at the very least that these two parts are definitely not made to work together. I suspect that someone tried to repair your window before and used the wrong hardware. I highly recommend checking to see if your house has an identical window that still uses all the original hardware. Without seeing what the original setup was, it will be difficult to come up with a solution.