appears that both coils are broken

Thanks for posting. This looks like an older version of our 15-177 Pivot Shoe. This shoe works with Series 152 Coil Balances. Let me know if you have further questions about any of this hardware. 

To clarify, the older version isn't made anymore, so the product I recommended previously is our best and only option. It should replace the original without issue.

There should be a stamp printed on the band of the spring, near where it screws into the shoe. If you don't see it, or if it's too faded to read, then you will instead need to weigh the sash on a scale (any common bathroom scale will do) and then divide that weight by how many springs support it in total. 

For example, if your sash is 20 pounds and you have two springs per side, then you want to order four springs rated to carry 5 pounds.