We present near-infrared integral field spectroscopy of the central kiloparsec of 17 nearby luminous and ultra-luminous infrared galaxies undergoing major mergers. These observations were taken with OSIRIS assisted by the Keck I and II Adaptive Optics systems, providing spatial resolutions of a few tens of parsecs. The resulting kinematic maps reveal gas disks in at least 16 out of 19 nuclei and stellar disks in 11 out of 11 nuclei observed in these galaxy merger systems. In our late-stages mergers, these disks are young (stellar ages <30 Myr) and likely formed as gas disks that became unstable to star formation during the merger. On average, these disks have effective radii of a few hundred parsecs, masses between 108 and 1010 M ⊙, and v/σ between 1 and 5. These disks are similar to those created in high-resolution hydrodynamical simulations of gas-rich galaxy mergers, and favor short coalescence times for binary black holes. The few galaxies in our sample in earlier stages of mergers have disks that are larger (r eff ~ 200-1800 pc) and are likely remnants of the galactic disks that have not yet been completely disrupted by the merger.