= Solution
For fixed $k$ and $\ell$, the line $-z/\kappa$ becomes increasingly negative, so each intersection approaches its branch’s left-hand vertical asymptote. Write $z_n=a_n+\varepsilon_n$, where $a_n=(n-1/2)\pi$ and $\varepsilon_n>0$. The eigenvalue equation gives
$$
\kappa\cot\varepsilon_n=a_n+\varepsilon_n.
$$
Since $\cot\varepsilon=\varepsilon^{-1}-\varepsilon/3+O(\varepsilon^3)$, it follows that $\varepsilon_n=\kappa/a_n+O(a_n^{-3})$. Thus
$$
\boxed{\lambda_n=\frac{(n-1/2)^2\pi^2}{\ell^2}+\frac{2k}{\ell}+O(n^{-2}),\qquad\lambda_n\sim\frac{n^2\pi^2}{\ell^2}.}
$$
The half-integer phase identifies the sharper high-frequency behaviour of these <Dirichlet-Robin eigenvalues on a finite interval>. The expansion holds with the Robin coefficient fixed; it is not a uniform expansion in $k\to\infty$.
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