  We carry out an analysis of the multibody decay cascade $\Lambda_b\to
\Lambda^*_J J/\psi \to pK^- J/\psi$ with the $ \Lambda^*_J$ resonance including
$\Lambda^*_{1520},\Lambda^*_{1600}$, and $\Lambda^*_{1800}$, and $J/\psi$
reconstructed by the lepton pair final state. Using the helicity amplitude
technique, we derive a compact form for the angular distributions for the decay
chain, from which one can extract various one-dimensional distributions. Using
the $\Lambda_b\to \Lambda_J^*$ form factors from lattice QCD and quark model,
we calculate the differential and integrated partial widths. Decay branching
fractions are found as $\mathcal{B}(\Lambda_b\to \Lambda^*_J(
pK)J/\psi(\ell^+\ell^-))=(1.35\pm0.28)\times 10^{-5}$. In addition, we also
explore forward-backward asymmetry, and various polarizations. Results in this
work will serve a calibration for the study of $b\to s \ell^+\ell^-$ decays in
$\Lambda_b$ decays in future and provide useful information towards the
understanding of the properties of the $\Lambda_J^*$ baryons.
