  We present new high angular resolution near-infrared spectroscopic
observations of the nuclear star cluster surrounding the Milky Way's central
supermassive black hole. Using the integral-field spectrograph OSIRIS on Keck
II behind the laser-guide-star adaptive optics system, this spectroscopic
survey enables us to separate early-type (young, 4-6 Myr) and late-type (old,
>1 Gyr) stars with a completeness of 50% down to K' = 15.5 mag, which
corresponds to ~10 \msun for the early-type stars. This work increases the
radial extent of reported OSIRIS/Keck measurements by more than a factor of 3
from 4" to 14" (0.16 pc to 0.56 pc), along the projected disk of young stars.
For our analysis, we implement a new method of completeness correction using a
combination of star-planting simulations and Bayesian inference. We assign
probabilities for the spectral type of every source detected in deep imaging
down to K' = 15.5 mag using information from spectra, simulations, number
counts, and the distribution of stars. The inferred radial surface-density
profiles, $\Sigma(R) \propto R^{-\Gamma}$, for the young stars and late-type
giants are consistent with earlier results ($\Gamma_{early} = 0.93 \pm 0.09$,
$\Gamma_{late} = 0.16 \pm 0.07$). The late-type surface-density profile is
approximately flat out to the edge of the survey. While the late-type stellar
luminosity function is consistent with the Galactic bulge, the
completeness-corrected luminosity function of the early-type stars has
significantly more young stars at faint magnitudes compared to previous surveys
with similar depth. This luminosity function indicates that the corresponding
mass function of the young stars is likely less top-heavy than that inferred
from previous surveys.
