We present a comprehensive analysis of a broad band spectral line survey of the Orion Kleinmann-Low nebula (Orion KL), one of the most chemically rich regions in the Galaxy, using the HIFI instrument on board the Herschel Space Observatory. This survey spans a frequency range from 480 to 1907 GHz at a resolution of 1.1 MHz. These observations thus encompass the largest spectral coverage ever obtained toward this high-mass star-forming region in the sub-mm with high spectral resolution, and include frequencies $>$ 1 THz where the Earths atmosphere prevents observations from the ground. In all, we detect emission from 39 molecules (79 isotopologues). Combining this dataset with ground based mm spectroscopy obtained with the IRAM 30 m telescope, we model the molecular emission from the mm to the far-IR using the XCLASS program which assumes local thermodynamic equilibrium (LTE). Several molecules are also modeled with the MADEX non-LTE code. Because of the wide frequency coverage, our models are constrained by transitions over an unprecedented range in excitation energy. A reduced $chi^{2}$ analysis indicates that models for most species reproduce the observed emission well. In particular, most complex organics are well fit by LTE implying gas densities are high ($>$10$^6$ cm$^{-3}$) and excitation temperatures and column densities are well constrained. Molecular abundances are computed using H$_{2}$ column densities also derived from the HIFI survey. The distribution of rotation temperatures, $T_{rm rot}$, for molecules detected toward the hot core is significantly wider than the compact ridge, plateau, and extended ridge $T_{rm rot}$ distributions, indicating the hot core has the most complex thermal structure.
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