Just because you don't believe it, doesn't make it false.
I've had my 3DS since December 2011 and battery life on it is still pretty much just as good as it used to be then... and it doesn't randomly lose all its charge in a day
while powered off like yours does. The only time this would happen, was when I kept it in sleep mode all day
with my DSTwo inserted which is a -massive- battery drain. That was the only reason why it was always flashing red by the end of the day, being in sleep mode the whole time.
Heck, I've had my Gameboy Advance SP since like, what, 2004? And it doesn't even come -close- to having the kind of problem your 3DS has. Battery life on my SP are just fine, and it's been sitting there powered off for weeks and rarely used, but battery in it isn't discharged.
Same for my DS Lite... had it since December 2006, and it doesn't even come close to what your 3DS is experiencing.
Finally, I would recommend you read up on what fallacies are, because even assuming what she said there was false, it does not in any way discredit anything else.
https://en.wikipedia.org/wiki/Argument_from_fallacy