  The cross sections of the processes $e^+ e^- \to K^+ K^- J/\psi$ and
$K_S^0K_S^0J/\psi$ are measured via initial state radiation at center-of-mass
energies between the threshold and 6.0~GeV using a data sample of 980~fb$^{-1}$
collected with the Belle detector on or near the $\Upsilon(nS)$ resonances,
where $n=$1, 2, ..., 5. The cross sections for $e^+ e^- \to K^+ K^- J/\psi$ are
at a few pb level and the average cross section for $e^+ e^- \to
K_S^0K_S^0J/\psi$ is $1.8\pm 0.6 (\rm stat.)\pm 0.3 (\rm syst.)$~pb between 4.4
and 5.2~GeV. All of them are consistent with previously published results with
improved precision. A search for resonant structures and associated
intermediate states in the cross section of the process $e^+ e^- \to K^+ K^-
J/\psi$ is performed.
