I built that kit a few years ago. It's simple, and works well.
The tuning aid is a wheatstone bridge with 50 ohm resistors on three of the legs, and the (tuned) antenna on the fourth leg. The LED is across the bridge. When the antenna is presenting a 50 ohm load, the bridge is balanced, with 50 ohms on each leg, and no current should flow through the LED, so the LED should go out. Realistically, it's close enough if the LED just dims and doesn't completely extinguish.
About the only thing I can think of that would go wrong is if one or more of your 50 ohm resistors isn't really 50 ohms, or if something is not soldered in correctly. Another possibility is that your antenna is never coming close to presenting a 50 ohm load to the bridge.
If you can short out the antenna leg of the bridge, the input should see 50 ohms in parallel with 100 ohms, or 33 ohms presented to the input. That's easy to check with a multimeter. If you could substitute a 50 ohm resistor for the antenna leg of the bridge, you should be able to check with a multimeter that the input resistance of the bridge is 50 ohms (each leg should be 100 ohms, both legs in parallel make 50 ohms). If those don't check out, check soldering, and maybe disconnect the resistors to check each one's individual resistance.
