  We study effects of nonmagnetic impurities in a spin-1/2 frustrated
triangular antiferromagnet with the aim of understanding the observed
broadening of $^{13}$C NMR lines in the organic spin liquid material
$\kappa$-(ET)$_2$Cu$_2$(CN)$_3$. For high temperatures down to $J/3$, we
calculate local susceptibility near a nonmagnetic impurity and near a grain
boundary for the nearest neighbor Heisenberg model in high temperature series
expansion. We find that the local susceptibility decays to the uniform one in
few lattice spacings, and for a low density of impurities we would not be able
to explain the line broadening present in the experiments already at elevated
temperatures. At low temperatures, we assume a gapless spin liquid with a Fermi
surface of spinons. We calculate the local susceptibility in the mean field and
also go beyond the mean field by Gutzwiller projection. The zero temperature
local susceptibility decays as a power law and oscillates at $2 k_F$. As in the
high temperature analysis we find that a low density of impurities is not able
to explain the observed broadening of the lines. We are thus led to conclude
that there is more disorder in the system. We find that a large density of
point-like disorder gives broadening that is consistent with the experiment
down to about 5K, but that below this temperature additional mechanism is
likely needed.
