  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, biharmonic and 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.
