Since the mother lives in Tennessee, the younger sister likely is also living in TN. If the younger sister is living in TN, TN courts likely have jurisdiction in deciding who may have custody of her. If so, the orders regarding the younger sister likely would have to be obtained from TN.
If the younger sister's father is still living and his parental rights have not been terminated, he likely will need to be involved in any proceeding regarding the younger sister's custody.
You likely should review the facts and options with an attorney located in the state where the younger sister is living.
There is a good website for Washington guardianship, listed below. There is a difference between legal custody and guardianship. Guardianship is rarely 'temporary' in the sense that you can just decide whenever you want to terminate a guardianship. Does the younger sister need this because she is a minor or is she an incapacitated adult? if a minor then some kind of custody would be easier and expires when the minor becomes a legal adult.
After you obtain custody / guardianship you can register her for school but that doesn't automatically grant her Washington resident status for her first year.
Depending on what you mean by "providing", you can provide for her medical needs without any of this process. Just pay the bills. If you mean add her to your insurance as a dependent then consider that because you are someone's guardian does not necessarily make them your dependent for issues such as federal tax credits or insurance.