I am a CASA and one of the main goals I had for my youth was helping her bond with her foster family. I researched what she and I could do together, to support her healing overall. I have a lot of ideas for you.
Bessel Van der Kolk talks about the importance of rhythmic, synchronized group activity for regulation. That can be horseback riding, frisbee, contra dance, ping pong, or whatever the family enjoys. But you can give them a range of ideas. I often played tennis w my CASA youth, because it is a regulating activity but also one we both enjoyed.
I think a lot of the remedial learning kids need is about fundamental things they have never been exposed to. Explain things that you might think are 'obvious.' After 6 years in the home, my youth did not know anyone might WANT her to come home from college for Thanksgiving... and she would never have assumed that. Telling her "your foster family might feel disappointed not to be able to see you" was totally new information for her.
I think you can interpret the kid to the family, and vice versa, helping them "get" each other and be more tolerant and empathetic of one another.