Warning: I am not a professional by any means and haven't done much sewing so this is for novices like me! This is just how I made the cover work and might help inspire others if they are wondering what to do.
Here's the cover on the newly cut mattress:
I decided to cut out the triangular excess that you can easily see on each side. I cut out the top and bottom panel, leaving the sides which make up the width and house the zipper. The zipper runs from the bottom right corner up and around to the bottom left corner.
I did the same on the other side and then cut the fabric at the corners (after the zipper). Then you just pull the sides tight, trim, and sew it all back together.
I turned the whole thing inside out, pinned it up and sewed it back together (the cat was "helping").
The bottom corner close up. Not perfect, but it works!
And the end result! Click here to see the post on how I made the sheets.