  We study the vector and axial-vector form factors of radiative kaon decay
within the framework of the gauged nonlocal effective chiral action from the
instanton vacuum, focusing on the effects of flavor SU(3) symmetry breaking.
The general tendency of the results are rather similar to those of radiative
pion decays: The nonlocal contributions make the results of the vector form
factor increased by about $20\,\%$, whereas they reduce those of the
axial-vector form factor by almost $30\,\%$. Suppressing the prefactors
consisting of the kaon mass and the pion decay constant, we scrutinize how the
kaon form factors undergo changes as the mass of the strange current quark is
varied. Those related to the vector and second axial-vector form factors tend
to decrease monotonically as the strange quark mass increases, whereas that for
the axial-vector form factor decreases. When $K\to e\nu\gamma$ decay is
considered, both the results of the vector and axial-vector form factors at the
zero momentum transfer are in good agreement with the experimental data. The
results are also compared with those from chiral perturbation theory to $p^6$
order.
