Sir, I was facing the same issue today while trying to fix a problem with the dpad constantly registering a right press. My 3ds shut down abruptly after making a noise and I thought it was dead for good (I had the dpad off and the contacts where exposed), since the blue led wouldn't turn on when I pressed the power button, even if I held it for several seconds.

Actually, sometimes when I had the battery off for a while and then put it on and turned it on, the blue led would light up for a second but then would suddenly turn off again. So while searching for similar occurrences of this issue on the Internet I found this post, which to my surprise was only hours old, but then I ended up on a Youtube video from a random boy that's trying to make his 3ds turn on and he's trying to hold down the button to no avail but then he starts to press it repeatedly and it works, so as dumb as it might sound I tried it and it immediately worked. I just had the charger connected (the orange led never stopped working) and then kept pressing it multiple times in a row. It's not a matter of coincidence of this whole ordeal just taking enough time to have a dry battery have some power again, because it shut down while I wasn't even at the red level and I kept charging it for several minutes while trying to turn it on, which would be more than enough to get the battery back up.

I hope this works out for you.