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 $lambda/D$ 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 HST/NICMOS F110W and F170M image archive of 114 nearby brown dwarfs (observed in 7 different programs). We discover no new companions but recover and refine astrometry of 18 previous imaging companions (two with multiple epochs). We confirm two previous kernel-phase detections and do not recover a number of other candidates. We present contrast curves showing that our pipeline is able to detect companions down to flux ratios of $sim10^2$ at half the classical diffraction limit. We then conduct a demographic study of our detections and detection limits using a Bayesian framework. When incorporating a limit on unresolved binaries (based on an RV survey) we find a companion frequency of 25% which, even with sensitivity to much tighter separation companions, is consistent with previous studies of substellar multiplicity. We find a slightly wider distribution centered at a tighter separation than previous studies with lower sensitivity to tight companions. We also find a steep mass-ratio power-law index which depends on the assumed field age of the sample.