Chosen Solution
I’ve had the system for over two months. I dropped the system once, and shortly after there was no volume to be heard. Nintendo couldn’t repair it either.
The first thing to try is a reset. its the least expensive, or invasive repair. Follow the guide for factory resetting your DS. If nintendo tried repairing it, this would have been the first thing they do. a couple problems can cause the issue. on opening the back, try testing if a straight current into the speakers make it hum quietly. no more than 5V DC should be enough to get a noise. if not, its likely an audioIC failure. (Damage to the circuit on the board.) the cost of repairing this is senseless, and it would be better to buy a new one.