UPDATE:

Mom found this online recently, when looking into my 2DS problem, and it sounds EXACTLY like what's happening to mine RN, and they even say how to fix it!

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 Had this happen to my son's 2DS today, and I was able to fix it. For clarity's sake, my issue was this: no matter if the 2DS was connected to the AC adapter or not, when the power button was pressed, the blue LED would light for a couple of seconds and then extinguish. If you listened closely, you could hear the speaker "pop" when the LED turned off.
 
 How did I fix it? Follow along:
 
 Remove the battery cover, battery;
 Remove the back case, and gently release the rear camera ribbon cable from the mother board.
 On the motherboard, there are two other ribbon cables (near center). For some reason, my research landed on many people that said if the contacts on these cables get misaligned (through dropping, whatnot), it will cause the LED activity that I mentioned above.
 Release these two cables, then re-insert, and ensure that each cable is in completely (and straight) before clamping down again.
 Connect the battery, and test your DS. The reason I don't recommend putting everything back together - my research also indicated that since these two cables are so finicky, it could take several times to get it "just right".
 Gently put everything back together.
 Happy gamer!
 Score 12
 COMMENTS:
 
 Thanks, even though the ribbons appeared to be in properly, I removed them and placed back in and that did the trick!
 
 07/07/2016 by Neil Davidson

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When dad asked me to help with something, I put my 2DS in sleep mode. I put it back in its soft case momentarily, like normal. When I pulled it back out, to resume playing Pokémon afterwards, my system was off.

 This had actually happened to me before (not that it's supposed to), but unlike when it happened in the past, this time my 2DS system can't be turned back on. Every time I try to turn it back on now, the power light briefly lights up, then goes back out within seconds. The screen doesn't light up at all, nor does it make any sounds with or without headphones in.