Identifying and specing an Inductor
This is NOT a "film inductor", it is a standard wirewound high-power inductor. A simple Google search says it is FDA1055-H-2R2M=P3 made by Murata, 


  "Fixed Inductors 2.2uH 4.8mOhms 15.5A +/-20%"


and available at Mouser (and many other places)

However, it is highly unlikely that the faulty inductor is the root cause of your problem. First, this is a 16-A power supply. It needs some qualification to fix it, it could be transistor problems, or snubber cicuits deteriorated, tantalum caps expire (one of caps, TC8, seems a bit burned out). Or it could be nothing wrong at all. The board seems to be some high-density laptop mainboard. It is very likely that some very large chip has developed an INTERNAL problem with +5V rail, and it would be impossible to fix it. 

As an exercise, you can try to identify the source of overload by looking at excessive power dissipation, the guide can be found here, at SE.