You do not even need the equation for the ellipse if you use the property that, for any point $(x,y)$ on the ellipse, the sum of the distances from $(x,y)$ to the foci is a constant. Since we know that $(20\sqrt2,0)$ is a point on the ellipse, we can find this constant:
$\lVert (20\sqrt 2, -10)\rVert+\lVert(20\sqrt 2, 10)\rVert=2\sqrt{ 800+100}=60$.
Now we know that the y-coordinate of the points of intersection for the line going through $(0,10)$ is $10$, so we just need the x-coordinate. So we have that $\lVert(x,10)-(0,10)\rVert + \lVert(x,10)-(0,-10)\rVert = 60\\ \lvert x \vert + \sqrt {x^2+400}=60\\ x^2+400=60^2-120\lvert x\rvert+x^2\\ \lvert x \rvert=\frac {3200}{120}=\frac {80} 3$
So the points of intersection are $(\frac {80} 3 , 10)$ and $(\frac{-80} 3,10)$. The same method will allow you to find the other two points, try it! (You could also appeal to symmetry, but it's good to work through the argument on your own as well.)