We present spectrophotometric observations of the [Ni II] 1.191 μm and [Fe II] 1.257 μm lines at position 10 of Fesen and Kirshner in the Crab Nebula. To our knowledge, this is the first detection of the 1.19 μm line in a Galactic supernova remnant. Both lines are located in spectral regions relatively free of atmospheric absorption and airglow emission, and no confusion with other astrophysical lines exists. This makes these lines excellent probes of warm, low-excitation interstellar environments. Adopting an electron temperature of 16,000 K and an electron density of 850 cm-3, based on optical studies of the gas, we derive an Ni/Fe ratio of 0.3. This ratio is about a factor of 6 higher than the solar ratio but is significantly smaller than the Ni/Fe ratio derived from the [Ni II] 0.7378 and [Fe II] 0.8617 μm lines which yield Ni/Fe ratios of 50-75 times solar. Both of these ratios can be reconciled by increasing the electron density by a factor of ∼ 10. The derived Ni/Fe ratio from optical lines is left unchanged, while the ratio computed from infrared lines is increased by more than a factor of 4. However, such high densities seem physically implausible given the constraints placed on the conditions in the filaments from other observed lines. It is more likely that the atomic data for either Ni or Fe or both are in error causing the infrared and optical abundance ratios to disagree.