  In a companion paper \cite{jon-fei}, we established asymptotic formulae for
the joint moments of derivatives of the characteristic polynomials of CUE
random matrices. The leading order coefficients of these asymptotic formulae
are expressed as partition sums of derivatives of determinants of Hankel
matrices involving I-Bessel functions, with column indices shifted by Young
diagrams. In this paper, we continue the study of these joint moments and
establish more properties for their leading order coefficients, including
structure theorems and recursive relations. We also build a connection to a
solution of the $\sigma$-Painlev\'{e} III$'$ equation. In the process, we give
recursive formulae for the Taylor coefficients of the Hankel determinants
formed from I-Bessel functions that appear and find differential equations that
these determinants satisfy. The approach we establish is applicable to
determinants of general Hankel matrices whose columns are shifted by Young
diagrams.
