  We have theoretically investigated the structure of spinning self-bound
droplets made of $^{41}$K-$^{87}$Rb Bose mixture by solving the
Gross-Pitaevskii equation including beyond-mean-field correction in the
Lee-Huang-Yang form. The structure and energetics of vortex formation in the
self-bound mixture have been elucidated, showing that the formation of linear
vortices in the heavier species is energetically favoured over other
configurations. A fake (partially filled) core develops as a consequence in the
other species, resulting in a hole which might be imaged in experiments. The
interplay between vortices and capillary waves, which are the two ways angular
momentum can be stored in a swirling superfluid, is studied in detail by
computing the relation between angular momentum and rotational frequency. The
results show intriguing similarities with the case of a prototypical
superfluid, i.e. $^4$He droplets when set into rotation. A two-branches curve
in the stability diagram, qualitatively similar to the one expected for
classical (incompressible and viscous) rotating liquid droplets, is obtained
when vortices are present in the droplets, while prolate (i.e. non
axi-symmetric) shapes are only permitted in vortex-free droplets.
