How to calculate the energy state correction for second order degenerated pertubation theory where the degenarcy is not lifted to first order?

In Sakurai, there is a question (problem 5.12) about degenerated perturbation theory where the degeneracy is not lifted in first order. One is supposed to find the energy correction in second order for the Hamiltonian
begin{equation}
H=H_0 +delta H =
begin{pmatrix}
E1 & 0 & a
0 & E1 & b
a* & b* & E3
end{pmatrix}
;
text{with} ;;
delta H =
begin{pmatrix}
0 & 0 & a
0 & 0 & b
a* & b* & 0
end{pmatrix}
end{equation}
I’ve found the energy corrections to be $E^{(2)}_1 = 0$, $E^{(2)}_2=-frac{|a|^2+|b|^2}{E_3-E_1}$ and $E^{(2)}_3 = frac{|a|^2+|b|^2}{E_3-E_1}$.

Now I’m interested in how the state correction in first and second order $|n^{
(1/2)}$> would look like. I did a bit of research and found a blog article about that topic.

https://acollectionofelectrons.wordpress.com/2017/08/02/second-order-degeneracy/

The author explains how to obtain the state corrections in degenerate perturbation theory if the degeneracy is lifted in first order. It is not apparent for me how to generalize this for the case where the degeneracy is not lifted to first order.

How do I calculate the state corrections up to second order for $H$ or in general ?