  Panel-based, kernel-split quadrature is currently one of the most efficient
methods available for accurate evaluation of singular and nearly singular layer
potentials in two dimensions. However, it can fail completely for the layer
potentials belonging to the modified Helmholtz, modified biharmonic and
modified Stokes equations. These equations depend on a parameter, denoted
$\alpha$, and kernel-split quadrature loses its accuracy rapidly when this
parameter grows beyond a certain threshold. This paper describes an algorithm
that remedies this problem, using per-target adaptive sampling of the source
geometry. The refinement is carried out through recursive bisection, with a
carefully selected rule set. This maintains accuracy for a wide range of the
parameter $\alpha$, at an increased cost that scales as $\log \alpha$. Using
this algorithm allows kernel-split quadrature to be both accurate and efficient
for a much wider range of problems than previously possible.
