If the LED requires 10 milliamps to light, I don't think the controller is shorted, it is probably just the 5v system drawing power..... Now if the LED requires 3 AMPS to light, yeah, you have a short.
If you were "David", would you say you tested the controller and found it satisfactory if you hadn't actually done so? I don't have a clue who "David" is, but most of the folks I know wouldn't put their name to it without actually testing it.
So what to do? You surely don't want to damage your battery, and your experience so far has not been exactly wonderful.
First off, make sure you have the correct polarity at the battery. Is the positive marked terminal really the positive voltage source? If that is correct, then what?
Maybe follow Alan's suggestion?
So then, do you have a toaster or a waffle iron, or a clothes iron or something with a high power heating element controlled by a bimetallic switch? And a voltmeter? If you do, you can jumper wire the heating element in series with the controller, and measure the voltage drop across the heating element when you apply power. If the controller is shorted, you will see battery voltage across the heating element, and if the controller is ok, the voltage across the element should be very low. The high power resistor of the heating element will keep the current to a few amps, even if the controller is a dead short.
Please feel free to ignore every thing I have said, my wife does it all the time. But please pay attention to polarity.
TTFN,
Dennis