How do we file a claim with the insured’s insurance company?

You should contact the driver's (vehicle owner's )insurance  company and inform it in writing that your daughter will be filing a personal injury claim.
When your daughter completes her medical treatment and is released by the doctor or is declared by the doctor to be permanent and stationary, which means having reached a point in her medical treatment where no further improvement is anticipated, she (or you if she is a minor) should obtain her medical bills, medical reports, and documentation of wage loss.  Her personal injury claim filed with the registered  vehicle owner's insurance company should include those items.
Compensation for the medical bills is straight reimbursement.  The medical reports will document the nature and extent of her injury and will be used to determine compensation for pain and suffering which is an amount in addition to the medical bills.  Compensation for wage loss is straight reimbursement.
If the case is settled with the driver's (registered owner of the vehicle's) insurance company, NO lawsuit is filed.
 If your daughter (or you if she is a minor) is dissatisfied with settlement offers from the auto insurance company, reject the settlement offers and file a lawsuit for negligence against the registered owner of the vehicle and driver (if they are not the same individual).
If the case is NOT settled with the vehicle owner's insurance company, the lawsuit for negligence must be filed prior to the expiration of the applicable statute of limitations or your daughter will lose her rights forever in the matter.
If your daughter is a minor, you will need to be appointed guardian ad litem to file a lawsuit on her behalf because a minor cannot file a lawsuit herself.
If your daughter is NOT a minor, she can file the lawsuit herself.

You certainly can contact the driver's insurer, but bear in mind that they do not need to pay you and, in fact, will not pay if they feel your case is weak; their obligation is to their insured, and they will defend her in court and/or pay on her behalf if she is forced to pay (e.g. is sued and loses) or of they feel that your case is strong enough, and the amount you are seeking reasonable enough, that they are better off paying voluntarily rather than spending the money to fight a lawsuit but still losing and paying anyway.
If they will not pay voluntarily, you will have to sue the driver: you have to sue the driver, not the insurer.
Liability, or the obligation to pay for injuries or damages, is based on fault, which in cases like this, typically means driving negligently or carelessly. If the driver was driving carefully and the accident was not her fault--for example, there was a mechanical fault or a tire blew out, which caused the car to flip; or she had to swerve suddenly to avoid a person or animal stumbling out into the road--then she (and her insurer) will not have to pay.