The stamp only tells us how strong the spring is, though to tell the truth I'm not familiar with "12KL" either. I am assuming you have a
S770-12 because this is the product you were posting under. I agree that it looks similar, so I didn't have any reason to think otherwise. Do you believe you have a different Series?
While some companies may sell the same hardware, not everything will be the same. Some balance types have been known to be produced by different manufacturers, as well. It can be a little chaotic, which is why we like to rely on hard information like pictures, measurements, and sash weights. Manufacturing codes may be unreliable, but measurements never lie.
Right now, it'd probably be best to get us the following information:
1) The weight of your sash, which can be determined by removing it from the window and placing it on a scale.
2) A picture of how your balance attaches to the window. The
S770-12 attaches with a screw, but I am not sure how yours attaches.
3) A picture of the shoe that the metal clip hooks onto. That will let us confirm that the
S770-12 hook will work the same way.