= Solution
The flavor content is $K^0=d\bar s$ and $\bar K^0=s\bar d$. A charged-current <box diagram> changes <strangeness> by two units. The two internal <quark> lines can contain any up-type flavors $u_i,u_j\in\{u,c,t\}$, connected by two charged $W$ propagators. One allowed topology is shown below; its crossed counterpart also contributes to the full mixing amplitude.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-52-kaon-box.png]
{title=Charged weak box diagram converting a neutral kaon into its antikaon}
Each charged-current vertex contains the appropriate <CKM matrix> element. The flavor sum has combinations $\lambda_i\lambda_j$, with $\lambda_i=V_{is}^*V_{id}$, multiplying <mass>-dependent loop functions. <CKM matrix> unitarity gives $\sum_i\lambda_i=0$, illustrating the GIM cancellation of flavor-independent loop terms. The diagram is at order $g^4$ and second order in the <weak interaction>.
Let $\Theta$ be the antiunitary <CPT> operator. It interchanges the neutral-<kaon> flavor states up to phases. For a Hermitian <Hamiltonian> invariant under <CPT>,
$$
\langle\Theta K^0|H'|\Theta K^0\rangle
=\langle K^0|\Theta^{-1}H'\Theta|K^0\rangle^*
=\langle K^0|H'|K^0\rangle,
$$
since the diagonal expectation is real. Therefore \b[<CPT> gives $R_{11}=R_{22}$.] More generally an effective decay <Hamiltonian> has $R=M-i\Gamma/2$, where both $M$ and $\Gamma$ are Hermitian. <CPT> gives $M_{11}=M_{22}$ and $\Gamma_{11}=\Gamma_{22}$, hence the same equality of the complex diagonal entries. It does not require $R_{12}=R_{21}$; that is a <CP> condition. Nor is $R_{21}=R_{12}^*$ true for the full decay matrix in general.
Choose the <CP> convention $CP|K^0\rangle=-|\bar K^0\rangle$ and $CP|\bar K^0\rangle=-|K^0\rangle$. Then <CP> acts as $-\sigma_x$ on the flavor basis. Invariance of $H'$ means $R=\sigma_xR\sigma_x$, and therefore
$$
\boxed{\text{CP invariance: }R_{12}=R_{21}.}
$$
With a different flavor-state phase convention this relation carries the corresponding phase factors; equality is the relation in the convention adopted here. For a Hermitian <mass matrix> it makes the off-diagonal element real.
For the requested production-time <mass eigenstates>, discard the absorptive part. From now on $R$ denotes the Hermitian part $(R_{\rm original}+R_{\rm original}^\dagger)/2$, so its entries have the form
$$
R=\begin{pmatrix}r_0&z\\z^*&r_0\end{pmatrix},\qquad r_0\in\mathbb R,\quad z\ne0.
$$
If the original matrix already was Hermitian no replacement is needed. The <neutral-kaon mass matrix diagonalization> gives <eigenvalues> $r_0\pm|z|$, since $(r_0-\lambda)^2-|z|^2=0$. Write $z=r e^{i\varphi}$, $r=|z|>0$. Use the <kaon mixing square-root branch convention>
$$
a=\sqrt{R_{12}}=\sqrt r\,e^{i\varphi/2},\qquad
b=\sqrt{R_{21}}=\sqrt r\,e^{-i\varphi/2},\qquad ab=r,
$$
continuously from the <CP>-conserving convention in which $z>0$. Independent unrelated root signs would interchange the <eigenvalue> labels. Direct multiplication gives $R(a,-b)^T=(r_0-r)(a,-b)^T$ and $R(a,b)^T=(r_0+r)(a,b)^T$. Assigning the larger <mass> to $K_L^0$ gives
$$
\boxed{|K_S^0\rangle=\frac{a|K^0\rangle-b|\bar K^0\rangle}{\sqrt{2r}},\qquad
|K_L^0\rangle=\frac{a|K^0\rangle+b|\bar K^0\rangle}{\sqrt{2r}},}
$$
with <mass> shifts $r_0-r$ and $r_0+r$ respectively. A common strong-interaction <mass> may simply be added to both. Their norms are one and their inner product is zero. A Hermitian <mass>-only calculation determines the lighter and heavier combinations; it does not determine their different lifetimes.
In the <CP>-conserving limit, the chosen lighter and heavier combinations are
$$
\boxed{|K_1^0\rangle=\frac{|K^0\rangle-|\bar K^0\rangle}{\sqrt2},\qquad
|K_2^0\rangle=\frac{|K^0\rangle+|\bar K^0\rangle}{\sqrt2}.}
$$
They are <CP eigenstates> with <eigenvalues> $+1$ and $-1$ in the convention above. The sign of the real off-diagonal entry and the flavor-state convention are chosen together so that the stated $K_1^0=K_S^0$ limit is the lower-<mass> state.
Changing to this <CP> basis gives
$$
\begin{aligned}
a|K^0\rangle-b|\bar K^0\rangle
&=\frac{a+b}{\sqrt2}|K_1^0\rangle+\frac{a-b}{\sqrt2}|K_2^0\rangle,\\
a|K^0\rangle+b|\bar K^0\rangle
&=\frac{a+b}{\sqrt2}|K_2^0\rangle+\frac{a-b}{\sqrt2}|K_1^0\rangle.
\end{aligned}
$$
Consequently, provided $a+b\ne0$, define
$$
\boxed{\epsilon=\frac{a-b}{a+b}
=\frac{\sqrt{R_{12}}-\sqrt{R_{21}}}{\sqrt{R_{12}}+\sqrt{R_{21}}}.}
$$
Absorbing the common phase of $a+b$ into the definitions of the <mass eigenstates> and normalizing now yields
$$
\boxed{|K_S^0\rangle=\frac{|K_1^0\rangle+\epsilon|K_2^0\rangle}{\sqrt{1+|\epsilon|^2}},\qquad
|K_L^0\rangle=\frac{|K_2^0\rangle+\epsilon|K_1^0\rangle}{\sqrt{1+|\epsilon|^2}}.}
$$
For this Hermitian approximation, $\epsilon=i\tan(\varphi/2)$ is purely imaginary, so $\langle K_S^0|K_L^0\rangle=(\epsilon+\epsilon^*)/(1+|\epsilon|^2)=0$. The phase convention matters for the <kaon CP mixing parameter>. Physical <kaon> decay eigenstates instead diagonalize the generally non-<Hermitian matrix> $M-i\Gamma/2$, for which the mixing parameter can have a real part and the eigenstates need not be <orthogonal>. The formula here is the requested <mass>-only result, not a calculation of that full decay dynamics.
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