  We generate inverse Compton scattered X-rays in both linear and nonlinear
regimes with a 250 MeV laser wakefield electron accelerator and plasma mirror
by retro-reflecting the unused drive laser light to scatter from the
accelerated electrons. We characterize the X-rays using a CsI(Tl) voxelated
scintillator that measures their total energy and divergence as a function of
plasma mirror distance from the accelerator exit. At each plasma mirror
position, these X-ray properties are correlated with the measured fluence and
inferred intensity of the laser pulse after driving the accelerator to
determine the laser strength parameter $a_0$. The results show that ICS X-rays
are generated at $a_0$ ranging from $0.3\pm0.1$ to $1.65\pm0.25$, and exceed
the strength of co-propagating bremsstrahlung and betatron X-rays at least
ten-fold throughout this range of $a_0$.
