  We extend a precise renorming result of Godefroy, Kalton, and Lancien
regarding asymptotically uniformly smooth norms of separable Banach spaces with
Szlenk index $\omega$. For every ordinal $\xi$, we characterize the operators,
and therefore the Banach spaces, which admit a $\xi$-asymptotically uniformly
smooth norm with power type modulus and compute for those operators the best
possible exponent in terms of the values of $Sz_\xi(\cdot, \varepsilon)$. We
also introduce the $\xi$-Szlenk power type and investigate ideal and
factorization properties of classes associated with the $\xi$-Szlenk power
type.
