As far as I know, "finalement" is more like "lastly" in English. So it would mean at the end, or to finish off. I don't think it could be used in this sentence, but someone else may know better.

You can use either enfin or finalement - Larousse and le Petit Robert, for example, both use the alternative in the other's definition, or list them as synonyms. Enfin is likely used more often - it is what my wife (French) would have used, and what I heard more often in France (but I haven't formally checked usage statistics anywhere). One reason some suggest avoiding the near-English sounding choice when translating is to avoid faux ami. However, if you are certain of the correct meaning, that is not a major issue.