Help needed using convolution to calculate a system's impulse response
I was hoping someone could have a look through what I've done so far,
and also help me with the final integration.

You don't need to perform that integration.
In the s-domain the unit impulse is unity hence, you have calculated the response already and did the inverse transform correctly. The answer is \$t\cdot e^{-t}\$. The great thing about transferring to the s-domain is that if you can also transfer the time domain input signal to the s-domain, you have a fairly simple calculation and there are on-line resources (such as this one) that can do the inverse manipulation for you.