The insurance company.
This holds true if you were to use your insurance company or pursue the property damage claim through the other party's carrier. The insurance company makes the determination regarding the cost of repairs if your vehicle is not totaled. They also determine whether or not your vehicle is a total loss.
There is no magic number to determine whether a car is repairable. In general, an insurance company will assess the damage. If those damages exceed 65-75% of the car's total value, it's probable they'll call the car a total loss. This percentage may vary from company to company and state to state.