Given parabola $y^2=16x$,

So focus will be $\left(4,0\right)$,

Now equation of focal chord passing through $\left(2,2\right)$ and focus $\left(4,0\right)$ will be $y=\frac{-2}{2}\left(x-4\right)\Rightarrow y+x=4 .......\left(1\right)$

Now given length of double ordinate is $24$,

So $2y=24\Rightarrow y=12$ putting in parabola equation we get,

${12}^2=16x\Rightarrow x=9$

Now putting the value of $x$ in equation $\left(1\right)$ we get $y=-5$

So the point of intersection is $\left(9,-5\right)$