Abdul-Qadir ibn Abi Salih Musa ibn Abdullah ibn Yahya ibn Muhammad ibn Abu Muhammad AbuBakr Dawud ibn Musa ibn Abdullah ibn Musa Jawni ibn Abdullah ibn Hassan al-Muthanna ibn Hassan ibn Ali ibn Abi Talib , and this on his father side so you can see why they call him al-Hassani due to his tracing up to the Prophet's Nephew Hassan ibn Ali. On his mother side, she is the daughter of a saintly person Abdullah Sawmai who was a direct descendant of Imam Husain ibn Ali making the Shaykh also al-Husayni due to this.
Al-Gilani was born in 1077 A.D. (470 A.H.) in a small town of Iranian Gilan Province. His ancestors were Syeds who settled in Gilan (arabacized to Jilan) hence the epiphet of al-Jilani.
“ Sayyid Abu Muhammad Abdul Qadir was born in Naif in the District of Gilan in Persia (Iran) in the month of Ramadan....His father's name was Abu Salih, a God-fearing man and a direct descendant of Hazrat Imam Hasan R.A., the eldest son of Ali R.A, the Holy Prophet's (SAW) first cousin, and of Fatima R.A his beloved daughter. His mother was the daughter of a saintly person- Abdullah Sawmai who was a direct descendant of Imam Husain A.S, the younger son of Ali R.A and Fatima R.A. Thus Sayyid Abdul Qadir was both a Hasani and Hussaini........ SUBHANALLAH