  We find an exact spherically symmetric magnetically charged black hole
solution to general relativity (GR) coupled to nonlinear electrodynamics (NED)
with an appropriate Lagrangian density. In turn, starting with this spherical
black hole as a seed metric, we construct a rotating spacetime, a modification
of Kerr black hole, using the revised Newman-Janis algorithm that depends on
mass, spin, and a NED parameter $g$. We find an exact expression for
thermodynamic quantities of the black holes like the mass, Hawking temperature,
entropy, heat capacity, and free energy expressed in terms of horizon radius,
and they show significant deviations from the Kerr case owing to NED. We also
calculate analytical expressions for effective Komar mass and angular momentum
for the rotating black hole and demonstrate that the Komar conserved quantity
corresponding to the null Killing vector at horizon obeys
$\mathcal{K}_{\chi}=2S_+T_+.$ The radiating counterpart renders a
generalization of Carmeli's spacetime as well as Vaidya's spacetime in the
appropriate limits.
