Infrared observations of the dusty, massive Homunculus Nebula around the luminous blue variable $\eta$ Carinae are crucial to characterize the mass-loss history and help constrain the mechanisms leading to the Great Eruption. We present the 2.4 - 670 $\mu$m spectral energy distribution, constructed from legacy ISO observations and new spectroscopy obtained with the {\em{Herschel Space Observatory}}. Using radiative transfer modeling, we find that the two best-fit dust models yield compositions which are consistent with CNO-processed material, with iron, pyroxene and other metal-rich silicates, corundum, and magnesium-iron sulfide in common. Spherical corundum grains are supported by the good match to a narrow 20.2 $\mu$m feature. Our preferred model contains nitrides AlN and Si$_3$N$_4$ in low abundances. Dust masses range from 0.25 to 0.44 $M_\odot$ but $M_{\rm{tot}} \ge$ 45 $M_\odot$ in both cases due to an expected high Fe gas-to-dust ratio. The bulk of dust is within a 5$"$ $\times$ 7$"$ central region. An additional compact feature is detected at 390 $\mu$m. We obtain $L_{\rm{IR}}$ = 2.96 $\times$ 10$^6$ $L_\odot$, a 25\% decline from an average of mid-IR photometric levels observed in 1971-1977. This indicates a reduction in circumstellar extinction in conjunction with an increase in visual brightness, allowing 25-40\% of optical and UV radiation to escape from the central source. We also present an analysis of $^{12}$CO and $^{13}$CO $J = 5-4$ through $9-8$ lines, showing that the abundances are consistent with expectations for CNO-processed material. The [$^{12}$C~{\sc{ii}}] line is detected in absorption, which we suspect originates in foreground material at very low excitation temperatures.

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