Backlight coil replacement problem

It's difficult to say without knowing more about the damage caused to the coil. Generally speaking, coils rarely fail. However, if you say you can see it as damaged (can you post a picture?) then it certainly needs to be replaced.

The big question is why did the coil get damaged? Is it physical (dropped phone etc), electrical (high surge current) or water damage (corrosion or short circuit)? The root problem that caused this that may also have damaged the rest of the circuitry. I suggest you test the other components with a multimeter and replace only those that need to be changed. The less heat you apply to logic board the better.

This is not a trivial repair given the dimensions and density of the components involved. Unless you are a very experienced DIY'er, I would suggest you consult with a qualified repair shop that does micro soldering.