You will need some tools that most people who tinker with electricity will have.
Step 1: WARNING
For example, if you burn yourself with the soldering iron that is your fault. If you get lead poisoning from eating the lead solder this is you fault. If you burn down your house because you did not insulate your repair properly this is your fault. If your spouse whacks you with the rolling pin because you destroyed his/her Christmas lights guess what - your fault.
Step 2: Tools and Materials Required
Voltage Detector (see picture for example)
Heat Shrink Tube
Step 3: Circuit Explanation
Another factor to consider is the current limit of an LED. LED's do not like a lot of current so besides the LED there is also a current limiting resistor in each light. For this repair we are actually going to bypass the faulty light. This will increase the current that goes through the remaining lights because we have eliminated part of the resistance. However, taking one light out should not increase the current enough to damage the remaining LED's in same circuit. There is of course a limit. Bypassing any more that two lights in the same series circuit will likely increase the current enough to put the remaining LED lights in jeopardy which will definitely destroy the entire half of the light string.
Step 4: Detection
With the lights plugged in, use the voltage detector to check the live voltage wire between each light, starting at the end that plugs into the wall outlet. There should be voltage detected at the live wire going into the first light. Then check the wire that goes from the first light to second light. This is assuming that the first half of the light string is not working. If it is the second then start at light 36 for a 70 light string.
Keep checking for voltage between the lights. As soon as you find a wire between the lights that no longer has voltage detected the light prior is likely the light that is fault. As an example, if there is voltage on the wire between light 6 and light 7, but there is not voltage between light 7 and light 8, then light 7 is likely the problem.
Just to make sure test for voltage between the next set of lights just to make sure. There should be no voltage detected.
Step 5: Verification - DO NOT ATTEMPT
When I was sure of the LED that was faulty I unplugged the lights from the electrical source and then put a sewing pin through the wires going into the LED to bypass the light in question.
After making sure the pin was not touching anything conductive I plugged the lights back in and the light string half that did not work before now worked except for the light I bypassed with the pin. This proved I found the faulty LED.
I then unplugged the lights and began the repair.
Step 6: Repair
Step 7: Conclusion
As I mentioned earlier the removal of a light will increase the current and LED's do not like high current.
I measure the current in the light string half that was not faulty and it was 9mA. Measuring the current in the light string half with the one light removed the current was 12mA. The general upper limit for LED's is 20mA so the current level is still reasonable even with the one light removed.
I am happy to say the lights are still working fine after two weeks.