I was on the NY subway yesterday when I noticed that a woman that had just gone off the train left a bag under her seat. Inside the bag was a brand new pair of expensive sunglasses. The woman was not Jewish. Am I halachically supposed to try to find her? I understand that it will be a kiddush Hashem if I do return it but technically she doesn’t know that a Jew found it…
As you write, it will be a kiddush Hashem to return it, and the Iggros Moshe extols the virtue of doing so (under particular circumstances).
However, under the circumstances it will be certainly be a major effort to find her, and the kiddush Hashem will only be possible if you personally seek her out.
Therefore, in this case it is fine to keep the find for yourself.