\begin{align*} \epsilon_{\mu\nu\alpha\beta}\epsilon^{\alpha\beta\gamma\sigma} \partial_\gamma\partial^\nu \frac{|x|}{(k\cdot x)^2} &=\epsilon_{\mu\nu\alpha\beta}\epsilon^{\alpha\beta\gamma\sigma} \partial_\gamma\partial^\nu \frac{|x|}{(k\cdot x)^2}\ &=\epsilon_{\mu\nu\alpha\beta}\epsilon^{\alpha\beta\gamma\sigma} \partial_\gamma \left( \left(\partial^\nu|x|\right)\frac{1}{(k\cdot x)^2} + |x|\partial^\nu\frac{1}{(k\cdot x)^2} \right)\ &=\epsilon_{\mu\nu\alpha\beta}\epsilon^{\alpha\beta\gamma\sigma} \partial_\gamma \left( \frac{x^\nu}{|x|(k\cdot x)^2} -2 |x|\frac{k^\nu}{(k\cdot x)^3} \right)\ &=\epsilon_{\mu\nu\alpha\beta}\epsilon^{\alpha\beta\gamma\sigma} \partial_\gamma \left( \frac{1}{(k\cdot x)^2}\left( \frac{x^\nu}{|x|} -2 |x|\frac{k^\nu}{k\cdot x} \right) \right)\ &=\epsilon_{\mu\nu\alpha\beta}\epsilon^{\alpha\beta\gamma\sigma} \left( \partial_\gamma \left( \frac{1}{(k\cdot x)^2} \right) \left( \frac{x^\nu}{|x|} -2 |x|\frac{k^\nu}{k\cdot x} \right) + \frac{1}{(k\cdot x)^2} \partial_\gamma \left( \frac{x^\nu}{|x|} -2 |x|\frac{k^\nu}{k\cdot x} \right)
\right)\ &=\epsilon_{\mu\nu\alpha\beta}\epsilon^{\alpha\beta\gamma\sigma} \left( -2\frac{k_\gamma}{(k\cdot x)^3} \left( \frac{x^\nu}{|x|} -2 |x|\frac{k^\nu}{k\cdot x} \right) + \frac{1}{(k\cdot x)^2} \partial_\gamma \left( \frac{x^\nu}{|x|} -2 |x|\frac{k^\nu}{k\cdot x} \right)
\right)\ &=\ldots \end{align*}