  We present a $\approx 11.5$ year adaptive optics (AO) study of stellar
variability and search for eclipsing binaries in the central $\sim 0.4$ pc
($\sim 10''$) of the Milky Way nuclear star cluster. We measure the photometry
of 563 stars using the Keck II NIRC2 imager ($K'$-band, $\lambda_0 = 2.124
\text{ } \mu \text{m}$). We achieve a photometric uncertainty floor of $\Delta
m_{K'} \sim 0.03$ ($\approx 3\%$), comparable to the highest precision achieved
in other AO studies. Approximately half of our sample ($50 \pm 2 \%$) shows
variability. $52 \pm 5\%$ of known early-type young stars and $43 \pm 4 \%$ of
known late-type giants are variable. These variability fractions are higher
than those of other young, massive star populations or late-type giants in
globular clusters, and can be largely explained by two factors. First, our
experiment time baseline is sensitive to long-term intrinsic stellar
variability. Second, the proper motion of stars behind spatial inhomogeneities
in the foreground extinction screen can lead to variability. We recover the two
known Galactic center eclipsing binary systems: IRS 16SW and S4-258 (E60). We
constrain the Galactic center eclipsing binary fraction of known early-type
stars to be at least $2.4 \pm 1.7\%$. We find no evidence of an eclipsing
binary among the young S-stars nor among the young stellar disk members. These
results are consistent with the local OB eclipsing binary fraction. We identify
a new periodic variable, S2-36, with a 39.43 day period. Further observations
are necessary to determine the nature of this source.
