Just got it, this was my second attempt. The first time I lost my first life where you have to hit the button to drop the box and use it to jump over the spinning blade, so a decent ways in. From there I made three more silly mistakes, just based not quite having the timing right or forgetting about one of those surprise jumps and falling. I screwed myself right before the end, though. I forgot that when you have to jump and switch the gravity that you actually have to press Square to activate the switch, for reasons unknown I thought if I just passed through it I'd be fine. No such luck, and I died twice before I realized my mistake. It probably wouldn't have worked out anyway, I hadn't played the game in a week and didn't practice the last four chapters beforehand, so I'm sure I would have died at least twice on those parts.
I tried a second time today and did fine. A few close calls and two deaths, but the end was easy since I played the last few chapters four or five times before starting a new game.