Yeah, time is pretty tight for a 2:13. Whenever you break out of your run to do a normal jump, try to jump as soon as possible after you stop running. That way, while you're jumping, you'll maintain most of your momentum from running. Also, you should hope to fall to the lower level immediately after stomping the fragile ground.
I managed to get this time again just now - my times between screens were 2:28, 2:23, 2:22, and 2:15. I guess target times don't help so much in such a short level, though.
I hope something here will help; if not, I can go into the fine details, since it's such a short level.