We present ∼400 km s-1 resolution profiles of the 17.94 and 25.99 μm [Fe II] transitions from SN 1987A at t ∼ 400 days after core collapse. These observations used the facility cooled grating spectrometer aboard NASA's Kuiper Airborne Observatory. The two profiles are similar and have FWHM line widths of ∼2700 km s-1. The higher signal-to-noise 18 μm profile is somewhat asymmetric, falling off more steeply on the redshifted side than on the blue. Gaussian fits to the profiles yield an average centroid velocity of 280 ± 140 km s-1 relative to the Large Magellanic Cloud. The wings of the profiles extend to velocities ≳ 3000 km s-1. This shows that a significant fraction of the iron has been mixed outward into the hydrogen-rich envelope, which has a minimum expansion velocity of 2100-2400 km s-1. Both profiles also contain an unresolved 3-5 σ emission feature on the redshifted wing at νLSR ∼ + 3900 km s-1. We interpret this feature as emission from a high-velocity clump of material containing ∼ 3% of the total iron mass. The total line flux of the 26 μm ground-state transition yields an optically thin, singly ionized iron mass of 0.026 M⊙, relatively independent of the assumed temperature. This is significantly less than the 0.06 M⊙ of Fe+ determined from the decline of the optical light curve and the ionization of measured nickel lines, implying that the iron transitions still have appreciable optical depth. However, because of the small change in the 26 μm line flux from our measurement at 250 days, and the similarity of our profiles to the 1.26 μm [Fe II] profile, most of the emission is believed to originate from optically thin material with a temperature of 4400 ± 400 K. A comparison of the data with spherically symmetric models indicates a power-law density exponent of -3.2 ± 1.1 and a minimum expansion velocity of 650 ± 650 km s-1 for this optically thin component. The [Fe II] line fluxes and profiles also imply that the remainder of the material has high optical depth and is distributed in clumps throughout the ejecta, rather than being concentrated at low velocities in the center of a smooth density distribution.