Here is a video to compliment dunnid's solution to show what you need to do.
At the beginning of the video you can see which door is the "Extending Platform" level in the Clocktower area.
I did the difficult jump first to reach the gems in the upper left of the level. You are still able to drop down from this location to get the other gems. You will most likely die several times before you make the jump, so this way you don't waste time collecting the other gems each time.
(sorry for the audio lag)