OP's question is about domestic travel, presumably LAX-Hawaii. There is no ID requirement for a 13 YOA so long as traveling with an "adult companion." No letters, no nothing.
From the TSA Website ---
TSA does not require children under 18 to provide identification when traveling with a companion within the United States. Contact the airline for questions regarding specific ID requirements for travelers under 18.
That said, the air carrier may want something and before consulting attorneys and drafting letters, OP ought to call the air carrier they will fly and ask whether the carrier requires anything of the 13 YOA.
All of the other suggestions relate to documents the OP should or may want to have in order to act as the child's guardian for school registration, medical treatment, benefits and the like. That is certainly something she should speak with an attorney about and has nothing to do with domestic travel.