From experience from working with Nintendo DS Lites, the reason that your DSi's screens light up for half a second and shut back off is likely due to a '''certain cable on the motherboard not being properly seated'''.
Since you mentioned that you took it apart, you may want to check if any of those ribbon cables are inserted all the way into their proper slots. Be careful, as those ZIF connectors are really delicate. The cables Nintendo puts into their DS'es are ''very'' sensitive if they are half a millimeter out of place. I'd wager it is one those super tiny ZIF connectors, like the touchscreen connector.
As for the square impression, was there originally something adhered on that chip, like a piece of foam or something else that cushions it? Or does it look burnt?