To get a nice solution it may be best to reconsider the assumption that $\lambda$ is normally distributed. The exponential model requires $\lambda>0$, and a $N(\mu,\sigma^2)$ distributed random value can be negative.

$$ f(t,s)= f(t\mid s)\cdot f(s) = \frac1{\theta^k\Gamma(k)} s e^{-st}\cdot s^{k-1}\cdot e^{-s/\theta} $$ $$ = \frac1{\theta^k\Gamma(k)} s^{k}\cdot e^{-st-s/\theta} $$ for $s>0$, $t>0$. Here $s=\lambda$.