Iridium oxides (iridates) provide good platform to study the complex interplay of spin-orbit coupling (SOC) interactions, correlation effects, Hund coupling and lattice degree of freedom. However, previous studies primarily focus on tetravalent (Ir4+, 5d5) and pentavalent (Ir5+, 5d4) iridates. Here, we turn our attention to a recently reported unprecedented trivalent (Ir3+, 5d6) iridates, K0.75Na0.25IrO2, crystalizes in a triangular lattice with edge-sharing IrO6 octahedra and alkali ions intercalated [IrO2]- layers. We theoretically determine the preferred occupied positions of the alkali ions from energetic viewpoints, and reproduce the experimentally observed insulating behavior and nonmagnetic (NM) properties. The SOC interactions play a critical role in the band dispersion, resulting in NM Jeff = 0 states. More intriguingly, our electronic structure not only confirms the experimental speculation of the presence of in-gap states and explains the abnormal low activation energy in K0.75Na0.25IrO, but also puts forward the in-gap states featured with nearly free electron characteristics. Our theoretical results provide new insights into the unconventional electronic structures of the trivalent iridates and imply promising applications in nanoelectronic devices such as ideal electron transport channels.