Cut an oval shape from the bottom or side of a cardboard box.
This is the bottom of the bed.
Fold the side of the bed along the oval shape and attach with lots of duct tape.
Cover the bottom and side of the bed with soft material.
(I bought a secondhand pillow and used the stuffing)
Tape it to the cardboard to stay in place.
Difficult to explain how i made the cover... Oval piece for the bottom, long rectangular piece for the side of the bed, sewn together, pinched here and there... guess you're all creative enough to figure out yourselves.
And maybe your cat can help, just as mine did.
Let the bed be tested after each step, it has to be purrfect, of course...
The bottom is uncoverd, so upside down you see cardboard. But this way it's easy to remove the fabric for washing, and no-one sees the under side...