Okay, so I have two planes which intersect at a right angle. I have their normal vectors (which have different lengths). They intersect and form a line. I now need to find the equations of these two planes which are at an angle of 45deg between the two other planes.

I tried normalizing the vectors, but that gives me ugly values. Any other way around?
If you have normal vectors with equal lenghts, then the angle you are looking for is the sum of those two vectors. (Diagonal of a rhombus in general) Your normal vectors don't have equal lenght, so normalize them (divide by the corresponding lenght) and then add. You started correctly.