If you want your vehicle repaired right away, you are almost certainly going to have to proceed without insisting the driver and/or car wash pay for the damage up front. Did you get the other driver's information? You stated you suspect the driver had on his brakes. Did you see brake lights? It's unlikely the car wash was designed so that a collision between two vehicles was possible during normal operation. However, it probably should not have allowed the driver to remain in his vehicle if he could not understand instructions.
Don't forget you have a duty to mitigate your own damages, so it's probably not a good idea to wait for the other driver and/or the car wash to take care of your damages. If you don't get your own carrier involved but repair right away, send a copy of the estimate with your proof of payment to the other driver and the car wash, and demand payment in full within a certain time period - maybe 30 days.