The main reason for connecting the USB cable was to try and evaluate what else is damaged.
The dimly lit 3.3v LED suggest more than the chip with a hole in it is damaged. The chip with the hole is responsible for turning Vin into 5v; connecting USB bypasses that.
So that the board has a bright 5V LED tells us 5v is okay with an external supply from USB. 3.3v is made from the 5v, but it doesn't work properly.
The only adittional information we can get from that is weather or not any of the remaining parts (3.3v regulator U2, processor, sd card or networking chip/module) is getting hot when powered from USB.
Beyond that the board will need to be repaired and trying to establish the extent of the damage will help evaluate if its worth while repairing. As it stands, it can quickly add up to a lot of work and parts.
Sorry to be the bearer of bad news. Still not clear on the cause of the failure, if it was bad wiring warranty wouldn't cover it but this particular failure has been a issue with some recent duets. If you add a picture showing the serial number and information on where and when it was purchased maybe the Duet people will call it a warranty replacement. If not You can ask the community for help, but without a clear diagnose that might not be an option.