Dirac Delta and Laplace Transform

Hi guys, my professor recently gave us this problem:

$\displaystyle y^{\prime\prime}+4y=\sum_{k=1}^{\infty}\delta(t-k\pi)$ with all zero initial conditions to solve using Laplace transforms. So I assume that even though it's an infinite sum, in this case you can take the Laplace transform of the summand and sum that to get...

$\displaystyle Y(s)=\frac{1}{s^2+4}\sum_{k=1}^{\infty}e^{-sk\pi}=\cfrac{1}{(e^{s\pi}-1)(s^2+4)}$.

However I'm pretty sure that isn't invertible, or if it is I can't see a way to invert it cleanly. Did I mess up somewhere? Thanks.