I finally beat that level last yesterday. It was the only one from the first 20 that I hadn't completed mostly because of that part. When I did beat it it was with 97 diamonds because I didn't jump right down the middle so I had to beat it again. After the checkpoint it gives you an opportunity to jump down from the very top but I knew I'd die more doing that than starting over. Now I have 3 more levels left in the ghost world before going back to complete most of the bonus levels.
I could not get all the diamonds jumping down without messing up my timing on the jump. What I did instead, was continue through the level until I get the checkpoint, and work my way up and to the top right, just following the normal level progression. Eventually you end up on that platform to the right of that drop, with the eyeballs on the left. I dropped to grab the diamonds I was missing, and then made my way back to the right. You do a little backtracking, but all the enemies are already cleared out so it's just simple platforming at that point. If you die though, you start at the checkpoint and all the enemies are back, so you have to kill them all over again. At least you don't have to deal with the first part of the level. Everything in that level, other than that one drop/hover to the sword dude is easy squeezie.