You just shared the experience that colleagues in my school share with me many times a year, asking the similar/same question - did I react ok, what should I do next?
Imo, you did very well, starting with connection that you made with the kid which made her trust you, in the first place.
It is good that was not big problem objectively speaking, meaning neither was about her or someone else.s health or life are in danger.
Main thing that I reply to my colleagues, although I am very glad that kids trust them to ask for help, is that they are taking responsibility on themselves and to be very wary; sometimes kids want us to keep as secret something very serious and that is just not smart/ethical/in line with our position is school.
On the other hand, we also have parents that are coming and telling something that we are obliged to inform other institutions, wanting to keep it a secret, and asking for some favor for their kid.