We demonstrate broadband tunability of light emission from dense (6,5) single-walled carbon nanotube thin films via efficient coupling to periodic arrays of gold nanodisks that support surface lattice resonances (SLRs). We thus eliminate the need to select single-walled carbon nanotubes (SWNTs) with different chiralities to obtain narrow linewidth emission at specific near-infrared wavelengths. Emission from these hybrid films is spectrally narrow (20-40 meV) yet broadly tunable (~1000-1500 nm) and highly directional (divergence < 1.5°). In addition, SLR scattering renders the emission highly polarized, even though the SWNTs are randomly distributed. Numerical simulations are applied to correlate the increased local electric fields around the nanodisks with the observed enhancement of directional emission. The ability to control the emission properties of a single type of near-infrared emitting SWNTs over a wide range of wavelengths will enable application of carbon nanotubes in multifunctional photonic devices.