Search for the judge's contact info though your county clerk's website (or google) and call the judge's judicial assistant (usually the judge's chambers main phone number) to ask where you should send the letter and whether the judge would accept such a letter.
This answer should not be construed as creating an attorney-client relationship, and is for informational purposes only.
You can write a letter to the judge but it is very unlikely he will dismiss the charge as that is up to the State Attorney(prosecutor) to decide. You should contact the prosecutor and see if he will cut you a break. You should make sure your Ins. has paid all damages and send proof of that payment to the prosecutor.
You can not just write a letter to a Judge and expect your ticket to be dismissed. What kind of citation did you receive? Was it a civil citation or a criminal citation. Either way, you will have to appear before a Judge and either have a hearing or trial if you want to even see if the Judge is willing to dismiss your case. If it was a criminal citation, the State Attorney can drop the charge as well. You really need to hire a lawyer if it is your desire to fight this citation appropriately.