You need to provide your insurance company with a report of the accident as soon as possible after the accident (if you haven't already), including the license plate, driver's license, and any other information you have on the driver/auto who hit you. The insurance companies will normally work out who pays for your property damage (car damage). You should get three estimates. Some carriers require that you take the car to repair shops they deal with. Contact your insurance company and ask them if they have done this and when you should expect to hear from the at fault driver's carrier. If no one is responsive you should speak with a lawyer in your community that handles auto accidents. They will normally give a free consultation to learn the facts of your case and decide whether to take it or tell you what to do. I do not practice in your state but a local attorney there will be able to tell you the statute of limitations. Don't waste time wondering about it - make an appointment and get information. Make the phone calls. You may not even understand your own insurance policy so take it with you and have the attorney look at it. Sometimes carriers deny coverage wrongfully.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
As I am licensed in Florida and Vermont, I cannot offer you specific legal advice as to Michigan state law. However, I must respectfully disagree with attorney Wilson above. You stated that you do not have collision damage coverage, so there is no point in trying to have your insurance company involved in getting your car fixed.
You should contact the adverse driver's insurance carrier and advise them of this accident. They will assign a claim number and an adjuster to view your car and estimate the damages. In the interim, I suggest that you get your own appraisal of the damages from an independent repair shop. That way you will know if the insurance adjuster's estimate is a fair one.
The adverse driver's insurance company is obligated to repair your vehicle or, if it is totaled, give you the fair market retail value of your auto, based upon its age, model and condition.
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You should always seek counsel face to face from an attorney who regularly practices auto accident law in Michigan because Michigan has a unique set of laws as it relates to automobile accidents.
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The most you can recover from the other driver (or their insurance company) is $500. That is statutory and often referred to as "mini-tort". The only exception is if the other driver was uninsured, then you can attempt to sue that driver personally for the entire amount of the repairs.
I have a sample letter you can use to send to their insurance company on my website at:
Good Luck to you,
/ Thomas James