Fixed nitrogen species generated by the early Earth's atmosphere are thought to be critical to the emergence of life and the sustenance of early metabolisms. A previous study estimated nitrogen fixation in the Hadean Earth's N2/CO2-dominated atmosphere; however, that previous study only considered a limited chemical network that produces NOx species (i.e., no HCN formation) via the thermochemical dissociation of N2 and CO2 in lightning flashes, followed by photochemistry. Here, we present an updated model of nitrogen fixation on Hadean Earth. We use the Chemical Equilibrium with Applications (CEA) thermochemical model to estimate lightning-induced NO and HCN formation and an updated version of KINETICS, the 1-D Caltech/JPL photochemical model, to assess the photochemical production of fixed nitrogen species that rain out into the Earth's early ocean. Our updated photochemical model contains hydrocarbon and nitrile chemistry, and we use a Geant4 simulation platform to consider nitrogen fixation stimulated by solar energetic particle deposition throughout the atmosphere. We study the impact of a novel reaction pathway for generating HCN via HCN2, inspired by the experimental results which suggest that reactions with CH radicals (from CH4 photolysis) may facilitate the incorporation of N into the molecular structure of aerosols. When the HCN2 reactions are added, we find that the HCN rainout rate rises by a factor of five in our 1-bar case and is about the same in our 2- and 12-bar cases. Finally, we estimate the equilibrium concentration of fixed nitrogen species under a kinetic steady state in the Hadean ocean, considering loss by hydrothermal vent circulation, photoreduction, and hydrolysis. These results inform our understanding of environments that may have been relevant to the formation of life on Earth, as well as processes that could lead to the emergence of life elsewhere in the universe.