We perform a systematic study of the $^{56}$Ni mass ($M_{rm Ni}$) of 27 stripped envelope supernovae (SESNe) by modeling their light-curve tails, highlighting that use of ``Arnetts rule overestimates $M_{rm Ni}$ for SESN by a factor of $sim$2. Recently, citet{Khatami2019} presented a new model relating the peak time ($t_{rm p}$) and luminosity ($L_{rm p}$) of a radioactive-powered SN to its $M_{rm Ni}$ that addresses several limitations of Arnett-like models, but depends on a dimensionless parameter, $beta$. Using observed $t_{rm p}$, $L_{rm p}$, and tail-measured $M_{rm Ni}$ values for 27 SESN, we observationally calibrate $beta$ for the first time. Despite scatter, we demonstrate that the model of citet{Khatami2019} with empirically-calibrated $beta$ values provides significantly improved measurements of $M_{rm Ni}$ when only photospheric data is available. However, these observationally-constrained $beta$ values are systematically lower than those inferred from numerical simulations, primarily because the observed sample has significantly higher (0.2-0.4 dex) $L_{rm p}$ for a given $M_{rm Ni}$. While effects due to composition, mixing, and asymmetry can increase $L_{rm p}$ current models cannot explain the systematically low $beta$ values. However, the discrepancy can be alleviated if $sim$7--50% of $L_{rm p}$ for the observed sample originates from sources other than $^{56}$Ni. Either shock cooling or magnetar spin-down could provide the requisite luminosity. Finally, we find that even with our improved measurements, the $M_{rm Ni}$ values of SESN are still a factor of $sim$3 larger than those of hydrogen-rich Type II SN, indicating that these supernovae are inherently different in terms of their progenitor initial mass distributions or explosion mechanisms.
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