  We present a calculation of the $\eta$-$\eta'$ mixing in the framework of
large-$N_c$ chiral perturbation theory. A general expression for the
$\eta$-$\eta'$ mixing at next-to-next-to-leading order (NNLO) is derived,
including higher-derivative terms up to fourth order in the four momentum,
kinetic and mass terms. In addition, the axial-vector decay constants of the
$\eta$-$\eta'$ system are determined at NNLO. The numerical analysis of the
results is performed successively at LO, NLO, and NNLO. We investigate the
influence of one-loop corrections, OZI-rule-violating parameters, and
$\mathcal{O}(N_c p^6)$ contact terms.
