We report calculations of the spin-orbit energy as a function of internuclear distance R within the b (3)Pi(u) state of K-2, and between the b (3)Pi(0u) and A (1)Sigma(u)(+) states, together with new spectroscopic data on the b state and previously unpublished data on the A state. Both the new data and previous data are fitted to Hamiltonian parameters using the discrete variable representation (DVR) method. The DVR matrix includes nonrelativistic Born-Oppenheimer potentials and spin-orbit interactions, which are scaled to match the known asymptotic limits and to best fit the experimental data. We report fitted Dunham coefficients that yield the A and b state potentials by means of the Rydberg-Klein-Rees method. These parameters thus take into account second-order spin-orbit perturbation shifts (from the vibrational levels of these two states) which are normally not considered in band-by-band fits to spectroscopic data.