When figuring out how to write a letter concerning Don John, we can reflect on what we know of Elizabethan society as well as Elizabethan vocabulary.
When writing a letter concerning Don John's punishment, we would need to assume that Don John is already in captivity and that the letter is between leaders to discuss his punishment. We could reflect on the Elizabethan court system in order to decide who should write the letter to whom.
In the Elizabethan court, Don John would be tried for treason and executed. In fact, Elizabethans viewing the play Much Ado About Nothing and hearing that Don John had earlier tried to overthrow his brother's rule but had been "recently reconciled to the prince [Don John's] brother," as we see Leonato explain, would have been horrified to learn that Don John had not been executed already (I.i.132-133). They would have known from their own history that failing to have Don John executed would only lead to further insurrections.
Also, in the Elizabethan court, Don John would have been tried in the criminal courts that heard the most serious crimes, which were the Courts of Assize. However, after the trial it would have been up to the monarch to decide on the punishment, or mode of execution; therefore, Prince Don Pedro would have decided on the punishment. Since the decision would have been up to Don Pedro, we can say that this letter could be written by Don Pedro to Governor Leonato, in order to inform the governor of the prince's decision concerning the prisoner who was found in Leonato's city and is probably still being imprisoned in his city.
There were many different forms of execution in Elizabethan society, and for your letter, you would need to decide which form of execution Don John should have, if you believe he should have any. The type of execution depended on class and crime. Most persons of State or noble birth would have been executed by beheading, even for a crime of treason, as we see with Mary, Queen of the Scots. Since Don John is Prince Don Pedro's illegitimate brother, we may be able to treat him as a nobleman. However, the crime of treason was also punishable by being hung, drawn, and quartered and this may also be applicable to Don John depending on whether or not you want to consider him a nobleman. You can view a description of these punishments to include in your prospective letter at this link.
Also, if you need to write your letter in Elizabethan language, it is possible to look up commonly used Elizabethan vocabulary words, which can be found in this link.