metal piece that goes into the sash and allows for the sash to connect onto the pivot shoe
Thanks for writing in to us! The metal piece you are searching for with the prongs is a wood window pivot bracket. Yours very closely resembles the 26-355 Pivot Bracket but you'll need to carefully look over the dimensions to determine if this is the right fit for your repairs.
The shoe you have here is the 15-104 Caradco Friction shoe. Look over the dimension and compare them to confirm that this is the replacement you need. If you find you need to replace the jambliner balance springs in your window, you'll need to contact a local millwork in your area to see if they might be able to get you set up with the replacements.