Is the lady that ran into the rear of your car responsible for my car if she did not have liability insurance?

Assuming the woman is At Fault and caused the accident, she is certainly responsible for the damage caused. Whether or not a person has insurance has no impact on liability and they are still legally required to compensate you for the damage. If she refuses to pay outright, you will have to take her to court in order to recover. That's the frustrating part when people don't carry insurance. In most states, drivers are required by law to carry liability insurance, so it is likely she was driving illegally. Whether a person has liability insurance or not has nothing to do with whether or not they are responsible for the damage they caused to another vehicle. If the other person is the one that was found to be the one that caused the damage, then they are liable for the damage, whether they have liability insurance or not.