  We calculate the kaonic contribution to the shear viscosity of quark matter
in the kaon-condensed color-flavor locked phase (CFL-K0). This contribution
comes from a light pseudo-Goldstone boson which arises from the spontaneous
breaking of the flavor symmetry by the kaon condensate. The other contribution,
from the exactly massless superfluid 'phonon", has been calculated previously.
We specialize to a particular form of the interaction lagrangian, parameterized
by a single coupling. We find that if we make reasonable guesses for the values
of the parameters of the effective theory, the kaons have a much smaller shear
viscosity than the superfluid phonons, but also a much shorter mean free path,
so they could easily provide the dominant contribution to the shear viscosity
of CFL-K0 quark matter in a neutron star in the temperature range 0.01 to 1 MeV
(10^8 to 10^10 K).
