We report the dielectric properties of a two-dimensional layered Ruddlesden-Popper halide perovskite ${\mathrm{Cs}}_{2}{\mathrm{PbI}}_{2}{\mathrm{Cl}}_{2}$ synthesized via a simple mechanochemical process to explore fundamental aspects of ionic conduction and relaxation mechanism over a wide temperature and frequency range. Several experimental techniques, such as complex impedance spectroscopy, alternating current (AC) conductivity spectroscopy, and complex electric modulus spectroscopy, have been employed to investigate the nuances of ionic conduction and relaxation mechanisms, and the results have been corroborated using different theoretical models, such as the Maxwell-Wagner equivalent circuit model, the modified Jonscher power law, the Havrilliak-Negami (HN), and the Kohlrausch-Williams-Watts (KWW) model. The contribution of the grains and grain boundaries to the total impedance in the system is estimated by the analysis of the Nyquist plots. In temperature-dependent AC conductivity spectra, a critical temperature (413 K) is observed, beyond which the conductivity increases abruptly. This critical temperature also defines two distinct temperature ranges: the low-temperature (303--413 K) and the high-temperature (423--463 K) regimes, where the ionic transport mechanism switches from the normal ionic transport to a vacancy-mediated ionic transport mechanism. A substantially high activation energy $\ensuremath{\sim}1.82\phantom{\rule{0.16em}{0ex}}(\ifmmode\pm\else\textpm\fi{}0.02)\phantom{\rule{0.28em}{0ex}}\mathrm{eV}$ is calculated from the Arrhenius plot of the ionic conductivity in the high-temperature region, while at the low-temperature region, the activation energy is found to be $\ensuremath{\sim}0.48\phantom{\rule{0.28em}{0ex}}(\ifmmode\pm\else\textpm\fi{}0.02)\phantom{\rule{0.28em}{0ex}}\mathrm{eV}$. The abrupt jump in the ionic conductivity beyond the critical temperature is attributed to the onset of the anionic vacancy-mediated enhanced ionic conductivity. Polaronic models have been used to interpret the AC conductivity and its power-law exponent. The activation energy obtained from ionic conductivity measurements is consistent with those calculated from relaxation time using the HN and KWW models. The presence of two master curves in time-temperature superposition scaling of AC conductivity and modulus loss spectra specifies the validity of two different conduction mechanisms.