Unlikely that the led would flatten the battery
I would be having a load test done on the battery as it may be discharging internally or give you the impression that it is flat because it will not produce the current needed to spin the starter
other pointers to check are the output from the alternator and if it is dc volts , voltage regulator , belt condition and tension
failing that put an amp meter in the circuit for the ground and see what current is flowing
Items of high current draw are bulbs ( glove box, boot, interior light), faulty diode to the alternator, fans, big amphilers left on and faulty ECM and light relays
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