Methodology
Given Assumption 1 and Approximation 1, the proportion of clients who have at least 1 service record with a valid SLK is:

Now:

so it follows that the total number of clients is:

To resolve this equation for Nt the values of PNi is required. These are unknown, given it is not possible to observe the whole population due to the records with invalid SLK values. This method imputes the unknown PNi using numerical methods, then uses these values to impute Nt.
The process starts with the distribution of number of records per client that were observed using the records with valid SLKs (PNi,SLK1). These values are then adjusted so that the following conditions are met.
Constraint 1
The sum of the imputed proportions is equal to 1. That is:

Constraint 2
The imputed proportion of clients with 1 service record is less than or equal to the observed equivalent proportion among clients with records with valid SLKs. That is:

This constraint is used because some of the clients observed to have only 1 record will, in fact, have additional records with invalid SLKs. It is unlikely that the true proportion of clients with 1 service record is higher than that observed using records with valid SLKs.
Constraint 3
The total number of service records that the imputed total number of clients and the imputed distribution of records per client imply is equal to the observed number of service records.
That is:

This constraint is used to ensure that the imputed values are consistent with the observed number of records.
Penalty function
Under Assumption 2 we want to limit how much the imputed proportions differ from the proportions observed via the records with valid SLK data. To achieve this we use a penalty function that increases as the distance between the imputed and observed proportions increases. this function is defined to be:

Using numerical methods, the P'N1, P'N2, ... P'N10 are chosen that the penalty function is minimised, subject to the 3 constraints.
The final step is to use the imputed proportions to calculate the imputed total number of clients:

The resulting number is then rounded to the nearest integer.