That is awful! I have never experienced the cast sort of thing, but I have a million ideas of ways to entertain her!
I have been a nanny forever, and I am really good at coming up with diversions:
Let's see, at 3 1/2 you could do things like blind fold her and have her guess what things are by smell. Things like chocolate, cinnamon, oranges, etc with strong odors.
You could read to her, or better yet have her tell you a story and write it down for her. This is a great one because years from now she can look back at it.
I know that a cast up to the rib cage prevents her from sitting, but you can get a triangular cushion from Bed Bath and Beyond for $15 that can help her use her arms while resting on her tummy. Then she can color, finger paint, play with dolls.
Again an idea that might cost a few dollars, but you could get one of the lights that sends colors up all over the ceiling, or starts. If you get the kind with the replaceable plastic inserts then you could also get plain plastic and have her color it. Then when it is on her ceiling she can see her own coloring rotate around the room.
Do you take her outside at all?
With a picnic blanket you could lay down with her and look at the clouds passing or play I-spy.
There are toys like the projector toys that you pull an arm down and it switches to the next slide (I can't for the life of me remember what they are called) or kaleidoscopes.
Let me know which of these inspire you the most and I will respond with more.
I wrote the above earlier, but I just thought of another idea! Her sister could also do puppet shows for her. This would probably be a great way for her sister to feel like she is able to help. They can make up the characters together and the Au Pair can help to make the puppets.