These 3D images from the Mayo Clinic show Katie's jaw anatomy much more clearly than anything we've ever seen before. Most of us have a vertical bone (the ramus), which extends from the base of the skull downward. Then, the jawbone itself is meant to attach to the bottom of the ramus, and extends horizontally so as to align with the upper jaw.
Katie is missing the ramus entirely. Instead, she has only the lower jawbone, which is rotated upward without anything to attach to. This is what causes her open mouth posture, and the small mouth that her tongue cannot fit into.
In the surgery this Thursday Dr. Matthews will fashion two ramus (vertical) bones from her own ribs, and will attach the new bone from the base of her skull to the end of her jawbone, forcing that lower jaw to rotate upward. It is doubtful that he'll be able to completely close her bite so that her front teeth meet, due to the limitations of the soft tissue (skin, muscle, ligaments), but he will be able to get some of her back molars to meet, allowing her to once again chew.