What should you do if your insurance already paid a hit and run claim and the driver who did the damage was discovered?

Let your insurance company know who the at-fault driver is, and let them initiate subrogation (a fancy term for: "Give us our money back."). If the at-fault driver is being charged, your carrier will likely turn over supporting documents to the D.A. assigned to the case, who will request restitution be paid (this should include your deductible, if one applied). What you can't do is sue the at-fault driver for all the damages and expect to keep your insurance settlement as well. It's one or the other.