We present the first L-band (2.8-4.1 μm) spectroscopy of κ Andromedae b, a ∼20MJup companion orbiting at 1' projected separation from its B9-type stellar host. We combine our Large Binocular Telescope (LBT) Arizona Lenslets for Exoplanet Spectroscopy (ALES) integral field spectrograph data with measurements from other instruments to analyze the atmosphere and physical characteristics of κ And b. We report a discrepancy of ∼20% (2σ) in the L' flux of κ And b when comparing to previously published values. We add an additional L' constraint using an unpublished imaging data set collected in 2013 using the LBT Interferometer/LMIRCam, the instrument in which the ALES module has been built.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 μm 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 .1 μm appear to be necessary. If the brighter L' photometry is accurate, there is a hint that subsolar metallicity may be required.