  Reionization of the cosmic neutral hydrogen by the first stars in the
Universe is an inhomogeneous process which produces spatial fluctuations in
free electron density. These fluctuations lead to observable signatures in
cosmological probes like the cosmic microwave background (CMB). We explore the
effect of the electron density fluctuations on CMB using photon-conserving
semi-numerical simulations of reionization named \texttt{SCRIPT}. We show that
the amplitude of the kinematic Sunyaev-Zeldovich (kSZ) and $B$-mode
polarization signal depends on the patchiness in the spatial distribution of
electrons along with the dependence on mid-point and extent of the reionization
history. Motivated by this finding, we provide new scaling relations for the
amplitude of kSZ and $B$-mode polarization signal which can capture the effects
arising from the mean optical depth, width of reionization, and spatial
fluctuations in the electron density arising from patchy reionization. We show
that the amplitude of the kSZ and $B$-mode polarization signal exhibits
different dependency on the width of reionization and the patchiness of
reionization, and hence a joint study of these CMB probes will be able to break
the degeneracy. By combining external datasets from 21~cm measurements, the
degeneracy can be further lifted by directly exploring the sizes of the ionized
regions.
