Well most of the time we use inductors in DC/DC convertors. A high current is switched ON and OFF through it (you can see on internet how a buck converter works for example). If your Inductor is hot, it may be your converter who is stuck or it may be because you have a short-circuit on the load so a high current goes through through the DC converter and then to the short-circuit.
If the light is off, it doesn't mean that it's not working anymore, it might mean that it has not enough power, which is normal because of a short-circuit or because the DC converter doesn't work.
Ther hot inductor doesn't mean that it's broken, some of them are designed for high current.
I think that there is a short-circuit somewhere on the load. You have to check with a multimeter where it can be.