It only poses an identity dilemma if you believe that one's DNA contains one's "soul" or some Aristotelian essence of who one really is. If you assume that one's brain, and hence one's mind, is a complex system formed of numerous interacting parts, then the fact that different parts may have different DNA isn't that big a deal.
Another thing to remember: one's mitochondria have entirely different DNA from one's regular DNA (and that's not to say anything of the numerous digestive bacteria living in symbiosis in one's gut).
Two cell lines but one mind, doesn't that pose an identity dilemna ? Is her brain also made of two different DNAs ? Big emergent questions behind that story!