The bulk viscosity $\zeta$ is a transport coefficient which is of central importance for various areas of modern physics. In particular, its determination for a fluid with two components is challenging, since it involves a complex interplay of multiple microscopic processes that operate on different time scales. Within the Chapman-Enskog framework, based on a series expansion of the Boltzmann distribution function, many previous works have derived the 1${}^{\mathrm{\text{st}}}$ order result for the $\zeta$ of a mixture. However, such a result fails to reproduce relevant physical features of the system, especially when the masses of the two components are similar. In this work we improve the 1${}^{\mathrm{\text{st}}}$ order Chapman-Enskog result by deriving the $\zeta$ at the 2${}^{\mathrm{\text{nd}}}$ order in the expansion. We show that this improved formula encodes many physical properties that the 1${}^{\mathrm{\text{st}}}$ order result misses: under specific conditions, the 2${}^{\mathrm{\text{nd}}}$ order result can be qualitatively and quantitatively very different from the 1${}^{\mathrm{\text{st}}}$ order one. Moreover, this result is compared against the $\zeta$ evaluated within the Green-Kubo formalism, by means of a numerical solution of the Relativistic Boltzmann equation. The agreement with respect to this benchmark is significantly improved when moving from the 1${}^{\mathrm{\text{st}}}$ to the 2${}^{\mathrm{\text{nd}}}$ order CE result.