...she would rid herself of the letter--perhaps before going back to Longbourn. In Chapter 58 in the book, after his second offer to Elizabeth is accepted, he says:

"I knew, that what I wrote must give you pain, but it was necessary. I hope you have destroyed the letter."

I think probably when Darcy wrote it, he did not think so much that he was sharing his sister's situation as he was focused on defending the charges against his character. I believe also, he thought he could depend on Elizabeth's secrecy because to disclose the information regarding Wickham, she might be pressed to disclose how she received that information, and why. I am sure she wanted no one to know of the "incident" which brought about the letter and would not spread the story.