  We present a systematic analysis of 191 stripped-envelope supernovae (SE
SNe), aimed to compute their $^{56}$Ni masses from the luminosity in their
radioactive tails ($M_\mathrm{Ni}^\mathrm{tail}$) and/or in their maximum
light, and the mean $^{56}$Ni and iron yields of SE SNe and core-collapse SNe.
Our sample consists of SNe IIb, Ib, and Ic from the literature and from the
Zwicky Transient Facility Bright Transient Survey. To calculate luminosities
from optical photometry, we compute bolometric corrections using 49 SE SNe with
optical and near-IR photometry, and develop corrections to account for the
unobserved UV and IR flux. We find that the equation of Khatami & Kasen for
radioactive $^{56}$Ni-powered transients with a single free parameter does not
fit the observed peak time-luminosity relation of SE SNe. Instead, we find a
correlation between $M_\mathrm{Ni}^\mathrm{tail}$, peak time, peak luminosity,
and decline rate, which allows measuring individual $^{56}$Ni masses to a
precision of 14%. Applying this method to the whole sample, we find, for SNe
IIb, Ib, and Ic, mean $^{56}$Ni masses of $0.066\pm0.006$, $0.082\pm0.009$, and
$0.132\pm0.011\,\mathrm{M}_{\odot}$, respectively. After accounting for their
relative rates, for SE SNe as a whole we compute mean $^{56}$Ni and iron yields
of $0.090\pm0.005$ and $0.097\pm0.007\,\mathrm{M}_{\odot}$, respectively.
Combining these results with the recent Type II SN mean $^{56}$Ni mass derived
by Rodr\'iguez et al., core-collapse SNe, as a whole, have mean $^{56}$Ni and
iron yields of $0.055\pm0.006$ and $0.058\pm0.007\,\mathrm{M}_{\odot}$,
respectively. We also find that radioactive $^{56}$Ni-powered models typically
underestimate the peak luminosity of SE SNe by 60-70%, suggesting the presence
of an additional power source contributing to the luminosity at peak.
