You can use both, I would use schießlich in this context, but it's close and you should probably not worry about it.
Schließlich (ultimately, in the end) is fitting here because it's about the conclusion of the process of searching in a third-person perspective.
Endlich (finally, at last) takes a personal perspective, it's always about a hope being finally fulfilled. It would be appropriate for a longer, more detailed account of her search from her view that draws the reader into the moment. In this context, endlich is less about conclusion of the search process and more about finding the person as a desired event that happens.
On the one hand, I think the description of the search is a bit too short to warrant endlich. On the other hand, you're already taking a subjective view by using erschütternd, so endlich doesn't seem out of place either.
You can find more about the difference between schließlich and endlich here:
What's the difference between "endlich" and "schließlich"?