As lollygag, said, she used the birth certificate, along with the court document for the name change, to get her passport. We don't know if they ever reissued a birth certificate under the name Lori Erica Kennedy. If they did, it probably would have also been in the box.
As for the note, we don't know if that note was actually in the box or if it was the only note of its kind. We have only been shown what the SSA investigator chose to share and what the Seattle Times decided to print.
i talked to my friend and he said in his case they did update the birth certificate but he didn't really need it as he had a court order showing his name change which, along with his new social security card, was the only documentation anyone official ever wanted. he did end up requesting the new birth certificate for completion's sake but said he's never used it for official documentation. results may vary etc etc but at least we know of one case where this is so.