Solving the equations of the given parabola and the line, we get,

${\left(\sqrt{2}y-4\sqrt{2}\right)}^2=4y\Rightarrow y^2-10y+16=0$

$\Rightarrow \left|y_1-y_2\right|=\sqrt{{\left(y_1+y_2\right)}^2-4y_1{y}_2}=\sqrt{100-64}=6$



$\Rightarrow$ Length of the chord $=AB=\sqrt{{\left(x_1-x_2\right)}^2+{\left(y_1-y_2\right)}^2}=\sqrt{2{\left(y_1-y_2\right)}^2+{\left(y_1-y_2\right)}^2}=\sqrt{3}\left|y_1-y_2\right|$

$=6\sqrt{3}$ units