My DSi goes blank after the health and safety message - How can I fix it?
With your power supply plugged in, do you see the central orange LED light up? If not, you have a power supply issue and no charge is getting to your battery. A severely depleted battery needs to reach a relatively high level of charge before the device will boot. Half an hour charge probably isn't even enough, so be sure to leave it for over an hour when dealing with very low battery levels.