We investigate the origin of the broadband visible emission in layered hybrid lead-halide perovskites and its connection with structural and photophysical properties. We study < 001 > oriented thin films of hexylammonium (HA) lead iodide, (C6H16N)(2)PbI4, and dodecylammonium (DA) lead iodide, (C12H28N)(2)Pb-4, by combining first-principles simulations with time-resolved photoluminescence, steady-state absorption and X-ray diffraction measurements on cooling from 300 to 4 K. Ultrafast transient absorption and photoluminescence measurements are used to track the formation and recombination of emissive states. In addition to the excitonic photoluminescence near the absorption edge, we find a red-shifted, broadband (full-width at half-maximum of about 0.4 eV), emission band below 200 K, similar to emission from < 110 > oriented bromide 2D perovskites at room temperature. The lifetime of this sub band-gap emission exceeds that of the excitonic transition by orders of magnitude. We use X-ray diffraction measurements to study the changes in crystal lattice with temperature. We report changes in the octahedral tilt and lattice spacing in both materials, together with a phase change around 200 K in DA(2)PbI(4). DFT simulations of the HA(2)PbI(4) crystal structure indicate that the low-energy emission is due to interstitial iodide and related Frenkel defects. Our results demonstrate that white-light emission is not limited to < 110 > oriented bromide 2D perovskites but a general property of this class of system, and highlight the importance of defect control for the formation of low-energy emissive sites, which can provide a pathway to design tailored white-light emitters.