In Chapter Twelve of The Egypt Game, why did the author describe the characters as being filled with helpless fuming anger?
Let us just briefly remind ourselves of what happens at the beginning of Chapter 12, entitled "Elizabethan Diplomacy." The four friends are in the yard and are surprised and terrified suddenly by a "shapeless inhuman figure" that jumps over the fence and leaps into the middle of the yard. It is only when Marshall notices a second figure climbing over the fence, dressed up in a costume that reveals the identity of the two mysterious arrivals, that they realise what is happening, and that they have been tricked by Toby and Ken, who have remained in their costumes to scare the four friends. This explains the reaction that your question refers to:
In all four Egyptians frozen fear boiled at once into a choking mixture of anger and relief.
They have realised that they are not being visited by some strange monster or demon, and thus on the one hand are intensely relieved, yet at the same time are filled with anger at how they have been tricked and scared by the boys.