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.
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