  We measure the gas-phase abundances of the elements He, N, O, Ne, S, Ar, and
Fe in the Lyman-continuum emitting region of the Sunburst Arc, a highly
magnified galaxy at redshift $z=2.37$. We detect the temperature-sensitive
auroral lines [SII]$\lambda\lambda4069,4076$, [OII]$\lambda\lambda7320,7330$,
[SIII]$\lambda6312$, [OIII]$\lambda4363$, and [NeIII]$\lambda3343$ in a stacked
spectrum of 5 multiple images of the Lyman-continuum emitter (LCE), from which
we directly measure the electron temperature in the low, intermediate, and high
ionization zones. We also detect the density-sensitive doublets of
[OII]$\lambda\lambda3727,3729$, [SII]$\lambda\lambda6717,6731$, and
[ArIV]$\lambda\lambda4713,4741$, which constrain the density in both the low-
and high-ionization gas. With these temperature and density measurements, we
measure gas-phase abundances with similar rigor as studies of local galaxies.
We measure a gas-phase metallicity for the LCE of
$12+\log(\textrm{O}/\textrm{H}) = 7.97 \pm 0.05$, and find an enhanced nitrogen
abundance $\log(\textrm{N}/\textrm{O}) = -0.65^{+0.16}_{-0.25}$. This nitrogen
abundance is consistent with enrichment from a population of Wolf-Rayet stars,
additional signatures of which are reported in a companion paper. Abundances of
sulfur, argon, neon, and iron are consistent with local low-metallicity HII
regions and low-redshift galaxies. This study represents the most complete
chemical abundance analysis of a galaxy at Cosmic Noon to date, which enables
direct comparisons between local HII regions and those in the distant universe.
