Here is how you can get a time of around 2:09, without using the double jump glitch. (I apologise if there are any English mistakes, it's not my native language)
Some important notes before starting : There is a constant timer going on, represented by the bar in the top left-hand corner of the screen. If it depletes completely, Haiku will die. When you die (whether because of the timer or other hazards), you are transported back to the starting room with all of your collected items. There is no way to refill the timer, except for three small orange power cells (or whatever they're called, haha) located in specific places that are also needed to beat the game. Two of them are on a path to get a mandatory item, and the last one you can only collect at the very end.
With that being said, here's the basic route for this run :
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First, go through the exit in the top left-hand corner (green) and collect the key, then immediately come back and take the exit on the opposite side of the starting room (grey). Go down to collect the little drone, then wait until you die.
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Go through the exit in the top right-hand corner (orange) and follow the path as normal until you get the wrench. It doesn't really matter how fast you go here, since you'll have to wait until you die again anyway.
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Go up through the green exit again, then follow the path. Once you get in the room that is green at the top and blue at the bottom, try to cut the corner as close as possible. After using the wrench, you can simply continue down and go to the right (the water can't kill you anymore). Collect the double jump item, then come back up (you have to really space out your two jump inputs, so be careful), go to the left this time, and kill yourself by walking in the water next to the robot. (this saves time over waiting until you die).
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Go through the grey exit and go back to where you found the drone at the start, then continue down the path and use your new double jump to reach the higher ledge and get the coin. Unfortunately you'll have to wait until you die here, and since you have more life than the first times, it will take a bit longer. Again, it doesn't really matter how fast you go for this part, as long as you get the item you need.
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Go back through the blue exit this time (bottom left-hand corner), then go to the bottom left again and give the coin to the robot (you don't have to wait or anything, just walk left continuously, careful to jump over the water though). Progress through the path until you get the scissors, then come back to the robot. There are some tricky jumps here that can make you lose time if you aren't careful, so I recommand watching a run and trying to get a feel for them. Once you are back in the room with the robot, walk into the water again to kill yourself.
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Go through the green exit, but this time use your double jump to get into the top left area. Progress through the path, use the scissors and get the last power cell. Go back to the starting room and through the grey exit, but this time take the upper path (the one that leads to the lab). You just have to wait until the power cells go into their respective places, and you've done it ! Congrats !