What will you do if the kid falls and hurts herself climbing the fence ? Will this be reason to finally go round to the parents and confront the issue?
Sorry to sound harsh, but I hadn't checked this thread in a while and I'm surprised the issue is ongoing. OP you are the authority figure here, the adult: this is a six-year-old child. If you don't want her in your house, tell her no. Firmly. Children need rules and boundaries. On the fence issue, speak to her parents. She could get hurt and this has to stop. According to your posts it seems like these people don't know what the hell is going on with their child. I don't know anyone who would let a child stay out until 8pm - hanging out all day in the house of someone who is pretty much a stranger to them.
If you don't feel comfortable speaking to child protection services, speak to the school.
You said the child called round after 6pm and you offered her food - she didn't ask. You are encouraging the problem by doing this.