How to handle delta function after finding the impulse response?
Since \$\delta(t) = 0\$ when \$t \ne 0\$, you should be able to replace:
$$f(t)\cdot\delta(t)$$ with $$f(0)\cdot\delta(t)$$
(I don't think this is always true in every case, e.g., inside of an integral, but I'm reasonably sure you can do it in your case)