Filling out the dearth of detections between direct imaging and radial velocity surveys will test theories of planet formation and (sub)stellar binarity across the full range of semimajor axes, connecting formation of close- to wide-separation gas giants and substellar companions. Direct detection of close-in companions is notoriously difficult: coronagraphs and point-spread function subtraction techniques fail near the {lambda}/D diffraction limit. We present a new faint companion detection pipeline called Argus that analyzes kernel phases, an interferometric observable analogous to closure phases from nonredundant aperture masking but utilizing the full unobscured telescope aperture. We demonstrate the pipeline, and the power of interferometry, by performing a companion search on the entire HST/NICMOS F110W and F170M image archive of 114 nearby brown dwarfs (observed in seven different programs). Our pipeline is able to detect companions down to flux ratios of ~102 at half the classical diffraction limit. We discover no new companions, but recover and refine astrometry of 19 previous imaging companions (two with multiple epochs) and confirm two previous kernel-phase detections. We discuss the limitations of this technique with respect to nondetections of previously confirmed or proposed companions. We present contrast curves to enable population studies to leverage nondetections and to demonstrate the strength of this technique at separations inaccessible to classical imaging techniques. The binary fraction of our sample ({epsilon}_b_=14.4_-3.0_^+4.7^%) is consistent with previous binary surveys, even with sensitivity to much tighter separation companions.