The problem we have to solve is cellular aging, especially with regards to Alzheimer's and other conditions. When DNA replicates from 5` to 3`, safety sequences of DNA known as telomeres get lost. This is a normal function of cell division, and causes aging, as when there are fewer telomeres available, the cell's "clock" keeps getting shorter.

The enzyme telomerase is responsible for keeping these in check. However, humans have very little telomerase.

The issue at hand is that if we present more telomerase, and thus extend the replication life of body cells, proofreading and repair mechanisms to repair illegal sequences caused by incorrect ligase bonding of Okazaki fragments may not be effective, as the statistical probability of errors increases. The other issue is that we don't know what can happen with an infinite telomerase supply. There could be ramifications on a molecular level that could be impossible to cure. Frameshifts, substitution errors -- there are an entire host of things to go wrong.

It's a long road to immortality. As no scientific data was published in the article, I'm not sure what exactly the girl has that causes her an inability to age. The only information given was obvious. Judging by her facial features, she seems like she has more of a growth disorder rather than a cellular stoppage of aging. If the latter were the case, she would still look very much like an infant, and would behave the same way as well.