Hybrid nanophotonic-electric networks-on-chip (NoC) have been recently proposed to overcome the challenges of high data transfer latencies and significant power dissipation in traditional electrical NoCs. But hybrid NoCs with nanophotonic guided waveguides and silicon microring resonator modulators impose many challenges such as high thermal tune up power, crossing losses, and high power dissipation. Due to these challenges productization of such architectures has yet to become commercially viable. Unfortunately, increasing embedded application complexity, hardware dependencies, and performance variability makes optimizing hybrid NoCs a daunting task because of the need to traverse a massive design space. No prior work has addressed the problem of synthesizing application-specific hybrid nanophotonic-electric NoCs with an irregular topology to the best of our knowledge. Considering the above unaddressed major challenges, in this paper we propose the HELIX framework for application-specific synthesis of hybrid NoC architectures that combine electrical NoCs with free-space nanophotonic NoCs. Based on our experimental studies, we demonstrate that the presented algorithms in this paper produce superior NoC architectures when compared to algorithms proposed in prior work for electrical NoCs.