Lily did not write a letter to Snape. Only the letter written by Lily to Sirius appeared in the book. (Part 7) After Dumbledore died and the power of the Fidelity Curse weakened, Snape went to Sirius's house. (Inherited by Harry) got the letter signed "Lily with infinite love" and a photo of Lily (actually a photo of the Potter family of three, which was torn in half by Snape).