We consider the regularity criteria of axisymmetric solutions to the non-resistive MHD system with non-zero swirl in ${\mathbb{R}}^3$. By applying a new anisotropic Hardy-Sobolev inequality in mixed Lorentz spaces, we show that strong solutions to this system can be smoothly extended beyond the possible blow-up time $T$ if the horizontal angular component of the velocity belongs to anisotropic Lorentz spaces.