Dont write a letter, its not allowed. You either hire an attorney to resolve case, go yourself or pay the ticket.
It is very inappropriate to write a letter to the judge and the judge will probably never see your letter anyway. Your argument that you live too far away/cannot afford to come to Detroit/etc. will not be accepted. Hire a MI attorney to handle this for you, or bite the bullet and make the trip.
You will receive more responses from Detroit area attorneys if you report using a Detroit location. You have targeted Illinois lawyers.
Writing a letter to the judge will not help you. If you are set on contesting the ticket, your best options are to either get yourself to court (check out Amtrak, Megabus, or Greyhound for affordable transportation from Chicago to Detroit) or contact a Detroit based attorney. Depending on the court, they may be able to handle the ticket without you present but it is best to ask a lawyer in the area about local rules. Many lawyers offer free consultations. Try giving them a call. From there you will be armed with information and ready to make the most educated decision you can. Best of luck!