An analytical solution.
The closed-loop system is $G(s)/(1+G(s))$ and its poles are those of $1+G(s)=0$.
In this case that is $$k \left(s^2+5 s+9\right)+(s+3) s^2=0 \ \ \ \ (1)$$.
For general third-order system with a pair of complex dominant poles, the poles are the roots of $(\alpha +s) \left(s^2 + 2 \zeta s \omega _n+\omega _n^2\right)=0$. Here $\alpha$ is the real pole, $\zeta$ is the damping factor, and $\omega _n$ is the natural frequency. In this case $\zeta=0.5$ and hence the equation becomes
$$(\alpha +s) \left(s^2 + s \omega _n+\omega _n^2\right)=0 \ \ \ \ (2)$$
If we equate the coefficients of (1) and (2) we will have 3 equations and 3 unknowns of which $k$ is one, and hence can be computed.
Expand[s^2*(s + 3) + k*(s^2 + 5*s + 9) - (s^2 + 2*(1/2)*wn *s + wn^2)*(s + a)]
eqns = Thread[CoefficientList[%, s] == 0]
sols = Solve[eqns, {a, k, wn}, Reals]
k /. sols
9 k + 5 k s + 3 s^2 - a s^2 + k s^2 - a s wn - s^2 wn - a wn^2 - s wn^2
{9 k - a wn^2 == 0, 5 k - a wn - wn^2 == 0, 3 - a + k - wn == 0}
{{a -> 3, k -> 0, wn -> 0}, {a -> 9/2, k -> 9/2, wn -> 3}}
{0, 9/2}
$k=0$ is an artifact of the calculations. The answer turns out to be $k=\frac{9}{2}$.