As you write the letter, keep in mind that Elizabeth is polite and respectful, so the letter should reflect that. She is also impressed by Wickham's kind and pleasant nature, in addition to his willingness to speak openly to her of his past. This makes her feel like a confidant, and like she is trusted and liked. The letter might go something like this; please look at mine as an example in order to write your own, with your own ideas and phrasing:
Dear Mr. Wickham,
I want to thank you for taking the time to visit with me and my sisters. We appreciated your courtesy and conversation. I am impressed by your easy-going manner, sense of humor, and your ability to speak frankly about deeply personal and hurtful matters; your story of Darcy and his callousness stirred me greatly. I am sorry that you have been treated so unkindly by a man of such stature, who should by right treat others with kindness and respect. I admire your decision to make your way in the world through joining the army, and I also admire your positive attitude. Many people would allow themselves to be overcome by their difficult circumstances, but somehow you have managed to stay pleasant, take things into your own hands, and keep bitterness out of your heart.
I look forward to further correspondance with you.