We present the first $L-$band (2.8 to 4.1~$mu$m) spectroscopy of $kappa$~Andromedae~b, a $sim20~M_{mathrm{Jup}}$ companion orbiting at $1^{primeprime}$ projected separation from its B9-type stellar host. We combine our Large Binocular Telescope ALES integral field spectrograph data with measurements from other instruments to analyze the atmosphere and physical characteristics of $kappa$~And~b. We report a discrepancy of $sim20%$ ($2sigma$) in the $L^{prime}$ flux of $kappa$~And~b when comparing to previously published values. We add an additional $L^{prime}$ constraint using an unpublished imaging dataset collected in 2013 using LBTI/LMIRCam, the instrument in which the ALES module has been built. The LMIRCam measurement is consistent with the ALES measurement, both suggesting a fainter $L$-band scaling than previous studies. The data, assuming the flux scaling measured by ALES and LMIRCam imaging, are well fit by an L3-type brown dwarf. Atmospheric model fits to measurements spanning 0.9-4.8~$mu$m reveal some tension with the predictions of evolutionary models, but the proper choice of cloud parameters can provide some relief. In particular, models with clouds extending to very-low pressures composed of grains $leq1~mu$m appear to be necessary. If the brighter $L^{prime}$ photometry is accurate, there is a hint that sub-solar metallicity may be required.
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