We develop a theory for the thermal Hall coefficient in a spin-$frac{1}{2}$ system on a strip of Kagome lattice, where a chiral spin-interaction term is present. To this end, we model the Kagome strip as a three-leg $XXZ$ spin-ladder, and use Bosonization to derive a low-energy theory for the spinons in this system. Introducing further a Dzyaloshinskii-Moriya interaction ($D$) and a tunable magnetic field ($B$), we identify three distinct $B$-dependent quantum phases: a valence-bond crystal (VBC), a metallic spin liquid (MSL) and a chiral spin liquid (CSL). In the presence of a temperature difference $Delta T$ between the top and the bottom edges of the strip, we evaluate the net heat current $J_h$ along the strip, and consequently the thermal Hall conductivity $kappa_{xy}$. We find that the VBC-MSL-CSL transitions are accompanied by a pronounced qualitative change in the behavior of $kappa_{xy}$ as a function of $B$. In particular, analogously to the quantum Hall effect, $kappa_{xy}$ in the CSL phase exhibits a quantized plateau centered around a commensurate value of the spinon filling factor $ u_spropto B/D$.
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