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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 semi-major 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 (PSF) subtraction techniques fail near the $λ/D$ diffraction limit. We present a new faint companion detection pipeline called Argus which analyzes kernel phases, an interferometric observable analogous to closure phases from non-redundant 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 \emph{HST/NICMOS} F110W and F170M image archive of 114 nearby brown dwarfs (observed in 7 different programs). Our pipeline is able to detect companions down to flux ratios of $\sim10^2$ 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 non-detections of previously confirmed or proposed companions. We present contrast curves to enable population studies to leverage non-detections and to demonstrate the strength of this technique at separations inaccessible to classical imaging techniques. The binary fraction of our sample ($ε_b=14.4^{+4.7}_{-3.0}$%) is consistent with previous binary surveys, even with sensitivity to much tighter separation companions.