\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*}