If you think about it, you cannot find the size of the square because, no matter what the size of the square, the angles remain the same. Different size squares would all be similar figures. When I said check the sides of the square, I only meant check that they were the same size as a final check that you had the right solution.
And, yes, apart from a small error in rounding one of the angles, you have the right answer! And the method works. Well done for persisting!