Can my attorney add that to the total settlement I won from the third-party which increased his fee?

That depends upon what your contract with your attorney says. Presumably, the settlement took into account that $5,000 in medical bills so it would be part of the settlement. In Northern California, contracts normally give the attorney a percentage of the net recovery after deduction of costs, but Southern Ca. attorneys charge on the basis of the gross recovery. Your attorney should have explained to you what his percentage basis would be.