The most common failure in computers IS the power supply, as to the LED then all this indicates is that the power supply is receiving power from the mains, not that the rest of it is working correctly.
Power supplies are cheap, just get one that is the same or more powerful then the one that is fitted, and also has the appropriate connectors for all your components. (Take the old one with you when you buy the new one)
In addition any computer shop worth its salt will have a power supply tester, and they will be able to tell you if your power supply is OK or not.
Hope this helps
Bill
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