We report on Gemini/GNIRS observations of the binary T Tauri system V582 Mon (KH 15D) at three orbital phases. These spectra allow us to untangle five components of the system: the photosphere and magnetosphere of star B, the jet, scattering properties of the ring material, and excess near-IR radiation previously attributed to a possible self-luminous planet. We confirm an early-K subgiant classification for star B and show that the magnetospheric He I emission line is variable, possibly indicating increased mass accretion at certain times. As expected, the H$_2$ emission features associated with the inner part of the jet show no variation with orbital phase. We show that the reflectance spectrum for the scattered light has a distinctive blue slope and spectral features consistent with scattering and absorption by a mixture of water and methane ice grains in the 1-50 $mu$m size range. This suggests that the methane frost line is closer than $sim$5 AU in this system, requiring that the grains be shielded from direct radiation. After correcting for features from the scattered light, jet, magnetosphere, and photosphere, we confirm the presence of leftover near-IR light from an additional source, detectable near minimum brightness. A spectral emission feature matching the model spectrum of a 10 M$_{J}$, 1 Myr old planet is found in the excess flux, but other expected features from this model are not seen. Our observations, therefore, tentatively support the picture that a luminous planet is present within the system, although they cannot yet be considered definitive.
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