It looks as though you have made a very good start on this problem. You do not actually need to show that the "square in the middle" (which I assume is the thing with the blue diagonals) is a square. It will be sufficient to show that it is a rhombus, because the diagonal of a rhombus intersect at right angles. To do that, prove that the red triangles are congruent (two sides, included angle).

In fact, it certainly seems to be true that the "square in the middle" really is a square, but I don't offhand see how to prove that.