  We study the scattering of a massless scalar field in a generic Kerr
background. Using a particular gauge choice based on the current conservation
of the radial equation, we give a generic formula for the scattering
coefficient in terms of the composite monodromy parameter $\sigma$ between the
inner and the outer horizons. Using the isomonodromy flow, we calculate
$\sigma$ exactly in terms of the Painlev\'e V $\tau$-function. We also show
that the eigenvalue problem for the angular equation (spheroidal harmonics) can
be calculated using the same techniques. We use recent developments relating
the Painlev\'e V $\tau$-function to Liouville irregular conformal blocks to
claim that this scattering problem is solved in the combinatorial sense, with
known expressions for the $\tau$-function near the critical points.
