That's a choke for power delivery, presumably for the battery (whose connector is right next to it).
If you're comfortable with soldering, you can try replacing the choke with an IDENTICAL choke; make sure you get the right one.
If you're not comfortable, then the only solution that I can think of would be a main logic board replacement. I don't think it's a problem with the battery, but rather the power delivery to and from the battery.