Based on experimental results, transient spark (TS) discharge in atmospheric air is an efficient source of NOx for biomedical applications with a negligible O3 production. The TS discharge is characterized by short (~10 ns) high current (~A) pulses initiated by streamer. The time-resolved optical imaging and spectroscopy of the TS discharge revealed that the primary streamer (ionization wave) is followed by the secondary streamer, enabling us to reconstruct the temporal evolution of the reduced electric field strength E/N(t). This was then used for a chemical kinetic model of the primary and the secondary streamer phases of the TS discharge. In this chemical kinetic modeling, we focus on the generation of selected reactive oxygen and nitrogen species (RONS) with antibacterial and other biological effects: O, N, NO, NO2, and O3. We proved that the secondary streamer plays a more important role in the induced chemistry than the primary streamer. In the simulation with the secondary streamer, the densities of RONS were increased by an order of magnitude when compared to the simulation without the secondary streamer, despite the same peak electron densities. The dominant intermediate product of the secondary streamer chemistry is atomic oxygen. Without the spark phase, this would lead to the generation of ozone as the dominant final product. However, in the spark pulse phase following the streamer(s), the chemistry is twisted towards dominant production of NOx.