My wife was recently involved in a car accident in a parking garage. We have full insurance, but she was driving her mother's car and the insurance on the car had lapsed without her mother realizing it. There is a question as to who was at fault in the accident. The other driver, who said she was not insured is trying to get us to pay for the damages to her car. Is my wife considered an insured driver even though she wasn't driving our car? And... will our liability insurance cover the accident?

Normally no, your wife would not be covered under another cars policy.

If the car she was driving was not insured, she was not insured. So if she is found at fault ( it may go to court) she and her mother will have to pay for the damgags