No, you're not responsible for helping to fix your friend's driving record. But as you were driving a vehicle that was uninsured, you are responsible for the damages caused to the party you hit and the damage to both vehicles.
Did you own a vehicle at the time. If so, your auto insurance may cover this loss. As far as your friend's record is concerned, you can't do anything about that. I would be more concerned that you may loose your license if the DMV learns your were driving without insurance (unless you had coverage of your own).
The first big question is whether you have your own insurance to cover the damage you have done? If you do have your own insurance, turn the claim over to your company and let them pay what needs to be paid. Do not worry about your friend's driving record. You cannot fix what you did not cause. I applaud your willingness to be accountable for what you did cause.