  We study the $\eta$-$\eta'$ mixing up to next-to-next-to-leading-order in
$U(3)$ chiral perturbation theory in the light of recent lattice simulations
and phenomenological inputs. A general treatment for the $\eta$-$\eta'$ mixing
at higher orders, with the higher-derivative, kinematic and mass mixing terms,
is addressed. The connections between the four mixing parameters in the
two-mixing-angle scheme and the low energy constants in the $U(3)$ chiral
effective theory are provided both for the singlet-octet and the quark-flavor
bases. The axial-vector decay constants of pion and kaon are studied in the
same order and confronted with the lattice simulation data as well. The
quark-mass dependences of $m_\eta$, $m_{\eta'}$ and $m_K$ are found to be well
described at next-to-leading order. Nonetheless, in order to simultaneously
describe the lattice data and phenomenological determinations for the
properties of light pseudoscalars $\pi, K, \eta$ and $\eta'$, the
next-to-next-to-leading order study is essential. Furthermore, the lattice and
phenomenological inputs are well reproduced for reasonable values of low the
energy constants, compatible with previous bibliography.
