Does the projector you purchased support keystone correction on both vertical and horizontal? The main problem I see is that your projecting your video “sideways”.
Have you tried placing a ruler against a wall and taking some throw ratio/height off-set measurements? I do this with every projector I purchase despite them all being the same model. Each can have a slight variance.
With these measurements, you can create a paper model of your projection, fold it to get a flat/keystone correct image on your surface and then build your projector and mirror mounts to match the angles required by your paper model.
It is also worth mentioning that most projectors only keystone correct vertically (the sides of the trapezoid). This means that you should be modeling the side view of the projection cone.
Hope this helps! Unfortnately, the_goat might be right about needing a different box.