  In the present paper, we prove convergence rates for the Local Discontinuous
Galerkin (LDG) approximation, proposed in Part I of the paper, for systems of
$p$-Navier-Stokes type and $p$-Stokes type with $p\in (2,\infty)$. The
convergence rates are optimal for linear ansatz functions. The results are
supported by numerical experiments.
