Wa`alykum As-Salaamu Warahmatullahi Wabarakaatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Dear brother, in Islam, first of all, let's express our appreciation for your forwarding this question to us, and we really commend your keenness in getting yourself well-acquainted with the teachings of Islam.
In the last third of Ramadan, a Muslim is recommended to increase his worship and devotion so as to bring himself nearer to Allah; he should make use of the last chance of the blessed month of Ramadan. Here, we recall the Hadith narrated by `Aishah, may Allah be pleased with her, in which she said that the Prophet, peace and blessings be upon him, used to observe I`tikaf during the last ten days of the month of Ramadan and he continued this practice until his death. After him his wives carried on this practice of observing I`tikaf regularly." (Reported by Al-Bukhari and Muslim)
As regards your question about "women performing I`tikaf in the mosque", the eminent Muslim scholar, Dr. `Abdul-Fattah `Ashoor, Professor of the Exegesis of the Qur'an at Al-Azhar University, states,
I`tikaf is recommended by the Sunnah for both men and women. The Prophet, peace and blessings be upon him, performed I`tikaf himself and so did his wives during his life and after his death.
Accordingly, there is nothing wrong if women perform I`tikaf in the mosque provided that this will not cause any negligence to the rights of their husbands or that of their children.
However, scholars of Al-Hanafiyyah and Al-Malikiyyah are of the opinion that women can perform I`tikaf in their homes where they are accustomed to performing prayer.
In conclusion, I think it is better for women to follow this latter opinion.
Allah Almighty knows best.