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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-massenergies 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 $e^+ e^- \to K^+ K^- J/\psi$ areat a few pb level and the average cross section for $e^+ e^- \toK_S^0K_S^0J/\psi$ is $1.8\pm 0.6 (\rm stat.)\pm 0.3 (\rm syst.)$~pb between 4.4and 5.2~GeV. All of them are consistent with previously published results withimproved precision. A search for resonant structures and associatedintermediate states in the cross section of the process $e^+ e^- \to K^+ K^-J/\psi$ is performed.

Updated Cross Section Measurement of $e^+ e^- \to K^+ K^- J/\psi$ and $K_S^0K_S^0J/\psi$ via Initial State Radiation at Belle