Families of confined rotating monopolar vortices are characterized using a variational formulation with the angular momentum as the driving force for confinement. The characterization for positive monopolar vortices given can be extended to negative vortices or to vortices within a rotating frame of reference. Besides the uniform Kirchhoff patches, new branches of vorticity solutions are found restricting the dynamics to level sets of both the angular momentum and the quadratic enstrophy. The rotation rate of the smooth vorticity structures depends on the vorticity profile. This is made perceptible by considering both minimum-energy vortices and maximizing vortices, rotating anticlockwise and clockwise, respectively. An approximation for the decay of the vortices due to dissipation is given in terms of the dissipation of vortices without loss of confinement. The elliptical Kirchhoff patches are found to symmetrize into circular patches. The minimum energy vortices gradually diminish while expanding their support, while the maximum energy vortices are unstable for the dissipative evolution.