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There definitive test for would be to test the voltage output from the battery terminals with a volt meter. It should output between 3.5-4vdc.
If you can't do that, then there is no other test than to replace the battery.
If the light is on when docked on the charger then power is getting through so it isn't the wall adapter or charger base.
You'd replace the battery first, and if that didn't help it would indicate the remote is at fault.
You can get a new battery here: http://amzn.to/1TX7sto