How do I write an apology letter to a teacher?

I lied to her. She doesn't know yet- but the guilt is eating me up. So I've decided that I'll write a letter to her. Problem is, I'm not quite sure how. She asked me if I had my essay 4 weeks after the due date, and I said that I had given it to her on time and we concluded that it must be lost. Truth be told- I didn't write the essay until 3 weeks after the due date, but I didn't tell her this because I did't want to risk being penalized for not doing the ignment on time... I do realize that now I'm in more trouble...How do I write this letter?