I was given permanent custody of one of my daughters friends who was a teenager. Now that she is 19 what is my legal relationship to her? Can she still list me as next of kin for medical issues.

I will say this is one I had never even thought of, since of course your legal status ended at 18, but since you were the legal guardian, I believe for legal purpose I would have them prepare a Limited Power of Attorney perhaps giving you the right only over medical issues in special cases.

This well needs to require an attorney

Yes - there's really no reason why she can't.