  We reinvestigate the structure of a steady axisymmetic force-free
magnetosphere around a Kerr black hole (BH). The BH magnetosphere structure is
governed by a second-order differential equation of $A_\phi$ depending on two
`free' functions $\Omega$ and $I$, where $A_\phi$ is the $\phi$ component of
the vector potential of the electromagnetic field, $\Omega$ is the angular
velocity of the magnetic field lines and $I$ is the poloidal electric current.
While the two functions $\Omega$ and $I$ are not arbitrarily given, which need
to be self-consistently determined along with the differential equation. Based
on the perturbation approach we proposed in paper I \citep{Pan2015a}, in this
paper, we self-consistently sort out two boundary conditions governing $\Omega$
and $I$, and interpret these conditions mathematically and physically. Making
use of the boundary conditions, we prove that all magnetic field lines crossing
the infinite-redshift surface also penetrate the event horizon. Furthermore, we
argue that the BH Meissner effect does not work in force-free magnetosphere due
to the perfect conductivity.
