Lattice QCD plays an increasing role in reducing theoretical uncertainties in kaon decays. Here we focus on kaon decay channels $K\to \ell\nu_\ell \ell'^+ \ell'^-$, which are closely related to radiative kaon decays since the lepton pairs $\ell'^+ \ell'^-$ come from virtual photon emission: $K\to \ell\nu_\ell \gamma^* \to \ell\nu_\ell \ell'^+ \ell'^-$. Compared with real photon emission, these channels involve more complicated form factors due to the off-shell photon with possible large momentum transfer, which causes a challenge to lattice-QCD studies. In this work, we introduce a lattice calculation procedure for their decay width, which can avoid parameterization of form factors. The systematic errors in our method are found to be controllable. Infinite volume reconstruction method is adopted to remove the temporal truncation effects and reduce finite volume effects from kaon intermediate state. This approach sets up a bridge between lattice QCD calculation and experimental measurement of decay width.