Typically it is better to go through an insurance company instead of dealing with a person one on one when there is substantial damage to a vehicle since the other person can renege on their deal and not pay.

If you contact the at-fault's insurance company it is likely they could find your daughter's car to be a total loss if the cost of repairing the car is the same or more than what the car is worth. This does not mean that you will not get paid though; the insurance company should have an obligation to 'make you whole'. Typically you should get a settlement for what the car was worth before being in the accident.

Negotiating An Insurance Claim Settlement is an article you might find of interest.