There are a few things you need to consider when checking the topology of your mesh. We've already pointed out the problems with normals and non-quad faces.
In the blend files you provided, I see that you already made the faces on the bottom-right of the mesh to quad but there are still a weird shading there :
The reason for this is duplicate geometry (edges and/or faces):
Entering face select mode makes the problem obvious, you probably created those face while attempting to fix the n-gon problem.
You can just delete those three faces and refill it :
With this edge selected press F and a face will be filled automatically. Same with those ones :
After that, the bottom part of the mesh should be clean :
There is another one here :
You can clean it the same way as the others.
For the upper part, just use the knife tool (pressing K) :
After that, you'll see that it's already better :
But you still have a triangle. Depending on what you want to do, you may want to fix it, you can't do this by either creating a loopcut to have an extra vertex or by merging the two edge loops to collapse the triangle, the two should be fairly easy to do.
Anyway, I hope this helped you as the question was how to get rid of the artifacts :