How do I properly express adding perturbed states to unperturbed states?

I have a problem set due tomorrow, and the last problem is driving me nuts. Been combing through griffiths trying to find similar examples to no avail, so it'd be greatly appreciated if stackexchange could help me out =)

Suppose that a quantum system has 4 independent states. The unperturbed energy of state $\lvert n_1\rangle$ is $E_1$ while the states $\lvert n_2\rangle$, $\lvert n_3\rangle$, and $\lvert n_4\rangle$ have energy $E_2$. Now a perturbation $H'$ acts on the system such that

\begin{align} H'\lvert n_1\rangle &= -k\lvert n_1\rangle \\ H'\lvert n_2\rangle &= k\lvert n_2\rangle \\ H'\lvert n_3\rangle &= k\sqrt{2}\lvert n_4\rangle \\ H'\lvert n_4\rangle &= k\sqrt{2}\lvert n_3\rangle + k\lvert n_4\rangle \end{align}

So the way I understand it, I have two unperturbed energy levels here, $E_2$ and $E_1$, the former of which is triply degenerate. When the system is perturbed, $E_2$ splits off into 3 new distinct energy levels. The first order correction would typically be given by

$$E_n = E_0 + \delta E_n,$$

where $E_n$ is the unperturbed Hamiltonian and $\delta E_n$ is the correction due to the perturbation. What I would have to do is add the perturbed states to the unperturbed states, but how do I work with the notation to express that mathematically? My base intuition is that I can treat the kets as eigenfunctions of $H^0$ and treat the $k$ coefficients as eigenvalues, but I'm not really sure if that would be going on the right track.