Use of Spiegeleisen in manufacture of steel
I didn't think Bessemer was used anymore. Spiegeleisen is more likely to be called ferro-manganese today. I expect it is added at the end to minimize losses of Mn to the slag. Mn does contribute to hardenability  but the primary reason for its use is to combine with residual sulfur to make MnS . MnS stringers can be harmful but are an improvement because otherwise the S will mostly be in grain boundaries reducing ductility and toughness. ( Bessemer process adds nitrogen to the steel reducing the toughness).  Calculated amount would be based on the total weight of Mn that must be added to get the desired content in the steel , such as 0.5 weight %.The added amount would need to allow for expected loss to slag, etc.