We theoretically study a topological nanospaser, which consists of a silver nanospheroid and MoS$_2$ monolayer flake of a circular shape. The metal nanospheroid acts as a plasmonic nanoresonator that supports two rotating modes, which are coupled to the corresponding valleys of MoS2. We apply external circularly polarized light that selectively pumps only one of the valleys of MoS$_2$. The generated spaser dynamics strongly depends on the size (radius) of the MoS$_2$ nanoflake. For a small radius, the system has only one spasing regime when only the chirally matched plasmon mode is generated, while at a larger size of MoS$_2$, depending on the pump intensity, there are two regimes. In one regime, only the chirally matched plasmon mode is generated, while in the other regime both chirally matched and chirally mismatched modes exist. Different regimes of spaser operation also have opposite handedness of the far-field radiated spaser system. Such a topological nanospaser has potential applications in different areas of infrared spectroscopy, sensing, probing, and biomedical treatment.