We report optical conductivity measurements of low resistivity LaNiO3 thin films grown on LaAlO3, (LaAlO3)(0.3) (Sr2AlTaO6)(0.7), and DyScO3 substrates, providing coherent compressive or tensile strain. The optical conductivity was measured by infrared reflectance and transmittance up to 3 eV and extrapolates well to the measured dc values. An unambiguous Drude response is clearly distinguishable from broad interband absorption extending beyond 3 eV. The measured spectral weight of the Drude response is substantially less than predicted by band theory, indicating an enhancement of the transport mass compared to the band mass by a strain-dependent factor of 2.5-5, characteristic of a strongly correlated metal.