People, as evidenced from this thread, really dont know what private fostering arrangements are
It is a private arrangement between two people, one who has PR, who is delegating the care of their child to a non relative (there are categories of this, but its too long winded to go into)
So the neighbour/friend etc has to agree that they are now going care for that child on behalf of the parent or person with PR
Have you done that OP
If you have agreed to this, then SSD will come to assess that to determine is not a Climbe situation, then you'll be left to get on with it more or less. There is a requirement to visit the child, DBS checks will be completed, an assessment of your capacity to meet need. The parent should change the child benefit over so you can make a claim, SSD will oversee it to some degree but its not an arrnagement made by them or set up by them, its not a 'placement'.
Private fostering, in my view, is best served when there has been a long standing 'parental' role from the private fosterer or they are an extremely significant person to the child
As someone points out above, you're not 'parenting' her where she is at yours, its game playing, staying over, hanging out. What would that look like as a permanent thing?