Are you using Zelda's teleport? I only failed that mission once when I teleported into the large spike above one of the bridges. As long as you know where you're going and aim the teleport properly, it should be fairly easy.

You can't jump your way up, Zelda is far too floaty and slow for that. By the time she lands from a jump you'll be at the bottom of the screen. Effective movement with Zelda is always about teleporting.