Flexible high-rise buildings with low damping are prone to excessive vibration under strong wind loads. To explore a light-weight control device having desirable mitigation effects on responses and sound robustness against deviations in tuning parameters, the performance of two novel inerter-integrated nonlinear energy sinks (NESIs), i.e., asymmetric nonlinear energy sink inerter (Asym NESI) and cubic NESI, on wind-induced vibration control of super high-rise buildings is assessed in the present work. Based on the wind loads obtained from wind tunnel tests, a super high-rise building with a 300 m height is taken as the host structure in the numerical case study. The results show that Asym NESI can achieve reduction ratios of 38.5% and 11.3% on extreme acceleration and displacement, respectively, while the sensitivity indices of Asym NESI on displacement and acceleration control are only 70.5% and 62.5% of those of tuned mass damper inerter (TMDI) having identical mitigation effects. Although the sensitivity indices of cubic NESI are only 5.5% and 29.8% of those of TMDI, the moderate mitigation effects and large nonlinear stiffness ratio may prohibit its practical implementation. Overall, Asym NESI could be an alternative to TMDI due to the same mitigation effects but better robustness against possible detuning.