Pull the power supply from the Xbox. Before you do it, unplug the Xbox and press the power button 2-3 times. There is a capacitor there that stores power. If you don't release that power by pressing the power button several times you can get shocked.
After you remove the power supply, look at the bottom. Right underneath the back are two prongs. These prongs may have some black around them. That means they are burnt. Sorry, I used to have pictures, but my hard drive crashed in the PC and I lost a lot of them. These prongs come loose from plugging and unplugging the power cable. If the cable it moved side to side to loosen it from the Xbox, these prongs get moved.
As stated before, clean the area really good. Rubbing alcohol usually works. Put your soldering iron on the prong and let it heat up. It usually helps if you have put some flux on there, but it isn't completely necessary. After the prong has heated, put your solder on the prong and let it flow. You are connecting the prong to the pad on the power supply. Do the same with the other prong.
If there aren't any burnt spots, you could have a fuse problem, depending on the power supply. When it is out, you should be able to read the name on the board.