I've been trying to find a way to add faces to the top of this mesh so that there are no double edges. Every attempt I have tried has failed. The Grid fill creates a maze of criss crossed lines. Bridge edge loops creates crossed faces. The knife tool refuses to cut through the top of the mesh and cuts down the side, (and I cannot make head nor tail of the constraints options for the knife tool at the bottom of the prompt bar). So I thought just joining 2 vertices would cut through any existing edges. It doesn't, it refuses to cut through. Only the Face tool 'F' works and of course, creates edges crossing over edges with no additional vertices. So how do I properly add faces to the top of this mesh so it is divided up with squares in each corner, a square in the middle and rectangles filling in the gaps, with no crossed over edges? I'm lost.
This might be a dumb question, but is there a reason you keep these two edges?
Because I cannot see a reason to keep them if you want to close the top, just select them and hit X > Delete > Edges:
Then select the top edge loop by holding Alt and clicking on one of the top edges:
Now just hit F and the top is completely filled with an n-gon. Afterwards you can select whatever vertices you want and join them with J to split the n-gon into multiple faces.