This page documents what happens when various components of your DSi have failed or are not connected properly. This should help if you've taken your DSi apart and find that it doesn't work upon re-assembling it.
Check that your fuse isn't blown - often caused by connecting the battery without the case on and shorting it.
Unit doesn't turn on, but blue light is on
Check that your Wifi module is connected properly.
Check that the small backlight ribbon for the bottom display is connected properly. Also check that the upper screen ribbon is connected properly.
Unit boots but channels won't launch (black screen, error has occurred)
Wifi module is not seated correctly or is damaged.
Unit hangs during boot up screen
Check that the camera ribbon is connected properly
Unit doesn't boot and yellow error code is displayed on top screen
Your NAND is corrupted. See Bootloader