For regular alkaline batteries, Nintendo says 60 hours lifetime using just the accelerometer, and 25 hours using both accelerometer and pointer. That's not very long for a kid that's seriously into a game.
Is it possible that you are also going through non-rechargeable batteries reasonably fast, but don't notice because, say, the kids can change those without your involvement, but with rechargeable batteries you are the one who handled recharging so saw every battery swap?
Another thing to watch out for is that alkaline and NiMH have very different discharge curves. Alkaline starts at 1.5, then falls steeply to around 1.3-1.4, then drops at a more gentle rate down to around 1.1, and then plummets.
NiMH starts at around 1.3, drops to 1.2, and then stays close to level until near the end, then starts dropping faster for a bit, then goes over a cliff.
If a device was not designed specifically to allow for NiMH, it is going to assume a discharge curve that follows the alkaline profile. When it uses voltage to estimate how much battery life is left, it is going to get confused by the NiMH curve.
If you put fresh NiMH batteries in a device, use it a while, and check the battery level, it will be reported as much lower then it really is. This can fool people into replacing the batteries early, because they figure it is falling fast and they don't have much time left.
I had one device where it would report not long after I put fresh batteries in that it was down to 60%, and if alkaline had fallen to 60% that fast, it would be time to put new batteries on my shopping list...but in reality with the NiMH it would actually stay at about 60% for weeks.