I am struggling to evaluate the following integral as part of a calculation
$$\int\limits\mathrm{d}\Omega_1\int\limits\mathrm{d}\Omega_2~Y^{*}_{1k}({\Omega_2})Y^{}_{1k}({\Omega_2}) \sum\limits_{l=0}^{\infty}\dfrac{4\pi}{2l+1} \sum\limits_{m=-l}^{l} Y^{*}_{lm}({\Omega_1})Y^{}_{lm}({\Omega_2}) $$
where $Y_{lm}(\Omega)$ are the spherical harmonics, $k = -1,0,1$ and the integration should be performed over the the unit sphere.
How can I use the orthogonal relation $$ \int\limits \mathrm{d}\Omega~Y_{lm}^*(\Omega)Y_{l'm'}^*(\Omega) = \delta_{l,l'}\delta_{m,m'}$$ and completness relation
$$ \sum\limits_{l=0}^{\infty} \sum\limits_{m=-l}^{l} Y^{*}_{lm}({\Omega_1})Y^{}_{lm}({\Omega_2}) = \delta(\varphi_1-\varphi_2)\cdot\delta(\cos\theta_1-\cos\theta_2)$$ to evaluate this?