You are dealing with a taxi cab company which is excellent putting barriers toward any settlement. The drivers are independent contractors and they carry their own insurance. Property damage is separate from bodily injury and should be settled separately. If after 9 months you have gotten them to repair your car, I would assume they are not going not unless you force the issue.
If you are not injured and it is only the car repair you want, I would sue the driver in small claims if the amount is below $5,000. If it is more than that you can sue in District Court though you will need to serve him personally. If you were injured as well, you can hire a personal injury attorney who can then include the property damage with the complaint for injuries.
An attorney in WA may be able to speed up the process.
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If you are seeking to have your repairs paid by your own insurance carrier, under a collision coverage provision, then 9 months is already too long. Your carrier has an obligation to resolve all first party claims within a reasonable [ = prompt] fashion. Call your carrier and tell them that if they do not settle your claim immediately, you will file a complaint with your State's Insurance Department. You can also retain an attorney to bring a lawsuit against your carrier for essentially what is a breach of contract claim. If the carrier you are referring to is for the taxi that hit you, don't waste anymore time waiting and just commence litigation.
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