What is the at fault driver responsible for?

Hello.  I am sorry to hear of your injuries.   To get the driver to pay for your bills, you will likely need to assert a claim with his/her insurance carrier.  If the insurance carrier denies the claim, you would need to file a lawsuit to recover.   There is no easy answer as to when the bills will get paid, as it depends in great deal on the specifics of your case, the nature of the injuries and the carrier response to the claim.  You may want to talk to a lawyer before filing a claim yourself. 

Unlike other types of cases, personal injury matters are usually handled on a contingency basis (meaning you pay no attorneys fees unless you recover some money) and many attorneys will provide a no-cost consultation in such cases.