To drive in Washington State a driver needs to be insured. This is why when taking the driving test for a driver's license proof of insurance is required. Most insurance companies will require that all licensed drivers in your household be listed and placed on your insurance policy since they are likely to drive your vehicles.
The reason an insurance carrier is so concerned about your daughter either remaining on the policy, turning in her license or even showing proof that your daughter now has her own insurance policy is because they may still be liable for damages if the driver is living in the house and is driving the car without having an insurance policy on her to drive it.
Most state's insurance laws allow an insurer to consider all resident operators of an insured vehicle in rating of an auto insurance policy. This includes your child, even if he or she has only a learner's permit or their first driver's license.
For information on what Washington State insurance laws regarding this situation check with the WA Office of the Insurance Commissioner. If you find that it is only this particular insurance company's guidelines that require you to keep insurance on your daughter and would like to find a company that may allow you to exclude her from your car insurance policy, click here for a WA State auto insurance quote.