Unfortunately, the symptoms that you are describing indicate that the internal power supply of the AirPort Extreme has failed.
Apple did not design the product for service, so repairs.....even if they were possible......would probably cost more than the original price of the AirPort Extreme.
If the AirPort is 5+ years old, there are other things that tend to start acting up as the device ages, so your best bet would be to retire the AirPort and replace it with another router.
Apple stopped offering the AirPort Extreme more that 3 years ago, but if you want to stay with an Apple product, there are usually some refurbished products available on Ebay or Amazon.
Otherwise, most users would do well to use a router that is offered by their Internet Service Provider since it will be basically preconfigured to work with your provider's service.