What is the correct solution for ridding oneself of the sin of taking a completely empty women's handbag found in the Prophet's Mosque, and is there an expiation for it to be performed there?

This issue is called "Luqatah" (found property). The ruling for found property in the Madinah Sanctuary is the same as elsewhere: it is permissible to take it and announce it for a full year. If its owner comes forward, then it is theirs; otherwise, the finder takes ownership of it. This is based on the hadith of Zayd ibn Khalid Al-Juhani (may Allah be pleased with him), who said: "Announce it for a year, then recognize its container and its tie. If someone comes who can describe it, then (it is theirs); otherwise, spend it."

However, since you are not a resident of the country and will return to your own, it was necessary to hand the bag over to the lost and found office for them to announce it. Since you traveled with it to your country, you must keep it until you return for Umrah or Hajj, then hand it over to the lost and found office, informing them of the date you found it.

It was obligatory for you to promptly announce it, as delaying it is a sin. However, we hope there is no sin upon you due to your ignorance of the ruling.