= Solution
Write $\kappa=\sqrt{2mV_0}/\hbar$ and $q=\kappa a$. At low <energy>, the $S$ wave normally dominates. Under the additional stated near-resonance assumption $ka\ll|\delta_0|$, neglect $ka$ inside the cotangent of the matching condition and set $k'\simeq\kappa$:
$$
k\cot\delta_0\simeq\kappa\cot q,\qquad
\sin^2\delta_0\simeq
\frac{k^2}{k^2+\kappa^2\cot^2q}.
$$
Substitution in $\sigma_0=4\pi\sin^2\delta_0/k^2$ gives
$$
\boxed{\sigma_{\rm res}\simeq
\frac{2\pi\hbar^2}{m}\,
\frac1{E+V_0\cot^2(\sqrt{2mV_0}\,a/\hbar)}.}
$$
This is a <near-resonance square-well approximation>, not a uniform formula for every well radius. A <phase shift> is defined modulo $\pi$; its representative must be chosen consistently when applying the comparison with $ka$.
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