You have no choice other than to send it to Nintendo for repair. You can either call their support at 1-800-255-3700 or go to this section of the website:
A fair warning, it WILL cost money to repair as water damage is caused by you and not a factory defect. I have never had a DSi repaired, but it can be assumed that the pricing is the same as a DS Lite that is no longer under warranty: $75.
Try leaving it alone for few days (maybe remove the battery). It may dry out and start working again. If after a few days, you still see the same symptoms, you will probably have to send it in.