Oh wow - I cannot believe you are still having to deal with this!
My first thought is she should have to EARN the toy in the first place verses getting it for 30 minutes and than being expected to rest ... A) many studies suggest that video games stimulate the mind of most people and actually 'impede' a restful state of mind so having the game first is counter productive to promoting sleep in someone who is struggling to 'rest' and B) she is likely spending that time focused on I am going to 'loose this' in x minutes anyway which is likely why she is starting to 'misbehave' anyway cause what does she have to loose she is going to get the thing taken away anyway?
Here the expectation is you rest quietly FIRST and if you do a good job you can earn a special quiet time activity ... if you do not REST quietly than you not only stay on your bed until the last person is up but you do not get to do anything 'special' for the rest of the day - you REST in a chair with books or your little imagination to keep you company and I make dang sure to plan something AWESOME ROCK YOUR WORLD FUN for the rest of the group!
I would also be talking to the parents with a WRITTEN WARNING and making it clear that rude disruptive behaviour at quiet time will NOT be tolerated in the program and while you do not care if she SLEEPS she MUST be quiet and respect the needs of both the children and yourself for quiet time and if this behaviour cannot be remedied within X days than your contract will have to be terminated ... and I would start advertizing for her space just in case the parents do not take the wake up call and put down the arm of the law at home with some consequences for her poor choices in CARE because there is only so much you can do within program with behaviour like this .... throwing a tantrum obviously is working SOMEWHERE for the child at this age or they would have learned long ago to behave!