Triad trigger: Whenever a long-bone fracture is followed by hypoxia, an altered mental state and petechiae, think fat embolism syndrome. Two of the three (lung and brain) are spelled out here.
Why fat embolism: Marrow fat enters the circulation through torn venous channels at the fracture site, plugs pulmonary and cerebral capillaries, and triggers an inflammatory cascade. That explains hypoxia unresponsive to high-flow oxygen and the confusion, classically a day or two after the injury.
Knocking out the others: A pulmonary contusion would light up the chest X-ray, but here the lung fields are clear. Hypovolaemic shock would show low blood pressure and tachycardia from blood loss, not pure refractory hypoxia with confusion. Thrombotic pulmonary embolism is uncommon in a child immediately after trauma and does not tie back to marrow fat. So the answer is fat embolism syndrome.