You need to open the system, check if the power supply is a foxlink. Foxlink power supplies have a bad solder job and you need to resolder the power plug so it will work again. (Reason being is because you have stated that the cord has been replaced, so therefore internal issue)
To Open the Xbox console, you will need a couple of special tools. You will need a TORX T-10 screwdriver and a Torx T-20 screwdriver. The XBOX case has 6 T-20 screws and to remove the hard drive and DVD drive, you will need to remove 3 torx T-10 screws. The power supply is held down by 2 torx T-10 screws.
http://www.llamma.com/xbox/Repairs/power_supply_failure.htm
The link above explains how to fix the issue.
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