= Solution
In each factor of the <BCS ground state>, the <BCS anomalous average> is $\langle b_k\rangle=u_kv_k=\Delta_k/(2E_k)$. Self-consistency therefore gives the zero-temperature <BCS gap equation>
$$
\boxed{\Delta_k=-\sum_{k'}V_{k,k'}\frac{\Delta_{k'}}{2\sqrt{\xi_{k'}^2+|\Delta_{k'}|^2}}}.
$$
Put $\Omega_D=\hbar\omega_D$. The specified constant attractive interaction makes $\Delta_k$ independent of $k$ inside the energy shell and zero outside it:
$$
\boxed{\Delta_k=\Delta\,\mathbf1_{\{|\xi_k|<\Omega_D\}}}.
$$
The printed $\delta_{k,0}$ is incompatible with this interaction: $k$ labels relative pair momentum, not the total momentum of a <Cooper pair>. Every pair here has zero total momentum, while many relative momenta contribute.
For the nonzero solution, the <constant-shell BCS gap equation> becomes $1=(V/L^3)\sum_{|\xi_k|<\Omega_D}(2\sqrt{\xi_k^2+\Delta^2})^{-1}$. Let $\nu_s$ denote the approximately constant <single-spin density of states> per unit volume at the <Fermi level>. Then
$$
1=\nu_sV\int_0^{\Omega_D}\frac{d\xi}{\sqrt{\xi^2+\Delta^2}}=\nu_sV\operatorname{arsinh}\frac{\Omega_D}{\Delta},\qquad \boxed{\Delta=\frac{\hbar\omega_D}{\sinh(1/(\nu_sV))}}.
$$
In <weak coupling>, $\Delta\simeq2\hbar\omega_De^{-1/(\nu_sV)}$. The displayed answer in the question uses $\nu=\nu_s$. If “total electronic density of states” includes both spin species, $\nu_{\mathrm{tot}}=2\nu_s$, the argument instead reads $2/(\nu_{\mathrm{tot}}V)$. If the density counts the whole box rather than unit volume, divide it by $L^3$ before using this formula.
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