You do not need the other driver's insurance to pay for a rental. It is the driver that is at fault and he/she has the contract with their insurance company, not you. They are ultimately responsible to make you whole, and that includes rental. so take care of yourself and then go after the driver if their insurance company does not play nice. you do not need to do anything with their insurance company. Their insurance company has a sole obligation to their client and of course to minimize the cash outlay in doing so.
I would contact the driver and let them know very clearly that they need to communicate with their insurance company to make sure you are taken care of otherwise they will find themselves in court.