  We develop a self-consistent approach for calculating the local impedance at
a rough surface of a chiral $p$-wave superconductor. Using the quasiclassical
Eilenberger-Larkin-Ovchinnikov formalism, we numerically find the pair
potential, pairing functions, and the surface density of states taking into
account diffusive electronic scattering at the surface. The obtained solutions
are then employed for studying the local complex conductivity and surface
impedance in the broad range of microwave frequencies (ranging from subgap to
above-gap values). We identify anomalous features of the surface impedance
caused by generation of odd-frequency superconductivity at the surface. The
results are compared with experimental data for Sr$_2$RuO$_4$ and provide a
microscopic explanation of the phenomenological two-fluid model suggested
earlier to explain anomalous features of the microwave response in this
material.
