State space representation in s-domain
From what I see what you did is correct.

I also couldn't think of a good way to do this without going to the time domain. Do note that from transfer function to state space, there are infinite possibilities. But some of the realizations are called canonical forms. See this wiki page for Canonical realizations.

Maybe what you can do is to get the complete transfer function for $$\frac{y}{u} = \frac{bk}{s^2 + (p+a)s + bk + pa}$$ first. And then just use that canonical transformation to plug in the numbers into the matrices by observation.