We cannot write your essay for you, but I will give you some ideas. By the time you would write such a letter to Jack, remember that he has gone over to the "dark side" - so your letter is going to have to convince him to return to his normal thinking, or the thinking he had before he went out of control.
I would focus on Jack's natural qualities and appeal to those things in the letter. You have learned that Jack is a natural leader and that he is good at getting others to follow him. One of the reasons he goes over to the dark side is that he engages in a power struggle with Ralph. When he can't get the rational ones to follow him, he appeals to the primitive instincts in the others and seduces them into following him. So your letter will need to reverse this action.
Dear Jack - I need your help! If we are all going to survive on this island, we need to work together. I realize that I cannot do this without you. You are a strong leader and the other boys look to you and respect you. You also are a good hunter. It's amazing how much you have learned about hunting since we have been on this island. We all need these skills to survive! You know that people are looking for us. It's just a matter of time until we are found. When the adults come for us, let's show them how grown up we are! Let's show them that we learned how to survive by ourselves. We can do this if we work together. Remember, you and I used to be friends. I miss working together with you.
You get the idea............you need to use psychology! Now, take it from there! For more ideas, see the analysis here on enotes.