I haven't mentioned it yet, so I'll mention it now:
My current theory is that Miyu had Dissociative Identity Disorder. The original Miyu was the primary personality / alter and New Miyu was (one of) her alter(s). When she fell from the roof, she sustained brain damage that erased the original Miyu's personality, leaving only New Miyu. If this is correct, it is able to explain how New Miyu is a totally different person that looks exactly the same and has the same name as the original Miyu without resorting to a supernatural explanation.
In nearly all real life cases of DID, alters do not share memories with each other, so it would make sense that New Miyu didn't know who the killer was. Original Miyu was in control of the body at that time.
Sure, we've seen a few scenes of both Miyus together, but this can easily be explained as a form of symbolism or metaphor.