If the power led comes on for a few second then powers off, then I think I know what the problem is.
On my ds lite once the top screen ribbon came loos and this happened, my advice to you is before buying a new Wifi unit, try to remove the wifi unit and see the connector for any dust ect , (if there is duct then simply blow at it) then replace it back on, do the same for the top screen and the bottom screen?
It can also be the small touch screen cable that has come loos ( this can also stop the unit to fully boot) NOT it is hard to open/close this small cable and the connector can easily brake. So I recommend you don?t do this as hopefully one of the other cable is at fault.
Also if you replaced the DS lite case recently and this happened then look for damages on ribbon cables of screens
