Enumeration of ideals of exceptional nilpotent matrix algebras

In well-known enumerations of characteristic ideals of the algebra $NT(n,K)$ of all (lower) niltriangular $n\times n$ matrices over a field $K$ and in related papers for nilpotent matrix groups and rings, the case $|K|=2$ is, as a rule, excluded from consideration; in this case, every ideal is characteristic. We find a formula for the number of all ideals of the algebra $NT(n,K)$ over any finite field $K$.