  Using a combination of time-domain THz spectroscopy (TDTS) and mutual
inductance measurements, we have investigated the low-energy electrodynamic
response of overdoped La$_{2-x}$Sr$_x$CuO$_4$ films that have been exposed to
ion irradiation. Films went through three rounds of irradiation (2, 4, and 6
$\times 10^{13}$ ions/cm$^2$) and mutual inductance and TDTS experiments were
performed between each step. Together with the as-grown film, this gives four
different levels of disorder. The transport scattering rate that is measured
directly in the THz experiments is an approximately linear function of the
radiation dose at all temperatures. This is consistent with a proportionate
increase in elastic scattering. In the superconducting state we find that the
relation between $T_c$, the superfluid density, and the scattering rates are
quantitatively at odds with the predictions based on the extant theory of
Abrikosov-Gorkov-like pair breaking in a dirty $d$-wave superconductor.
Increasing disorder causes only a small change in the superconducting
transition temperature for the overdoped films, but the changes to the $\omega
\sim 0$ superfluid density are much larger.
