Alright, I noticed what happened ^^
Even with that technique, I can't get over the gap. However, while recording a video for you to see, I had a small lag spike, and the character suddenly jumped way higher than it usually does.
Here's my theory : You seem to have made a mistake in the jump impulse. It doesn't take or incorrectly take into consideration the deltaTime between each frame. I have a pretty decent computer, and my framerate is pretty high. Due to that, it seems like the jump doesn't go high enough. With less framerate, I could jump higher.
Hopefully I'll be able to play a fixed version soon :-)