Your DSi shortcircuited what do you do?

2 Options available to you here.

1) If you are still within your store based warrenty time you can usually return the system to the store for a replacement. [This sometimes requires that you had purchased the extended warrenty, such as the warranty that FYE offers. This is what I have]

2) If option 1 is not available to you, your best bet is to Call the Nintendo support line and inform them of your problem. If the problem happened of no fault of your own and is a manufacturing defect, Nintendo usually will replace/repair the system at little to no cost to you. (with the exception of shipping the console to them.