All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
What you mentioned at the beginning of the question is a Qudsi (Sacred) hadeeth and not a verse from the Quran. It is a hadeeth reported by Al-Bukhari and Muslim and narrated by Abu Hurayrah, may Allah be pleased with him.
The second is a verse from Chapter As-Sajdah [Quran 32]. Allah says (what means): {And no soul knows what has been hidden for them of comfort for eyes as reward for what they used to do.} [Quran 32:17] This verse was mentioned within the text of the hadeeth as a confirmation of its meaning, as it is clear.
The hadeeth did not include a supplication; rather, it was mentioned as a statement.
Allah mentioned the reasons of attaining Jannatul-Firdaws (which is the highest, middle, and most beautiful part of Paradise) in general in His saying (which means): {Indeed, those who have believed and done righteous deeds - they will have the Gardens of Al-Firdaws as a lodging.} [Quran 18:107]
As-Sa’di said in his Tafseer (exegesis), “They are those who believed with their hearts and did good deeds with their body parts. This description included all of the religion: its beliefs, its deeds, its principles, and its outward and inward branches; then those people, despite the difference of their levels of faith and good deeds, for them are the gardens of Al-Firdaws.”
At the beginning of Soorah Al-Mu’minoon [Quran 23], some deeds are mentioned with regard to the reasons of entering Jannatul-Firdaws, and Allah ended them with the following; Allah says (what means): {Those are the inheritors. Who will inherit Al-Firdaws. They will abide therein eternally.} [Quran 23:10-11]
Martyrdom in the Cause of Allah and performing as much as possible of Nafl (supererogatory) prayers and acts of worship are among the reasons for entering Jannatul-Firdaws, but they are not the only reasons.
A Muslim should supplicate Allah to this effect [i.e. to enter Jannatul-Firdaws], so that Allah will facilitate this for him. Abu Hurayrah, may Allah be pleased with him, narrated that the Prophet said, “If you ask Allah, ask Him for Al-Firdaws, for it is the middle part and the highest part of Paradise.” [Al-Bukhari]
Asking Allah for Jannatul-Firdaws is included in the general meaning of the hadeeth that you mentioned, so if someone repeats it, then it is expected that he gets the virtues that are mentioned in it. In general, repeating the supplication to Allah is part of the etiquette of supplication; Ibn Mas’ood, may Allah be pleased with him, narrated that when the Prophet supplicated, he repeated the supplication three times. [Muslim]
Nonetheless, we are not aware of any specific supplication for asking Allah for Al-Firdaws, except what we have already mentioned above.
Also, we have not come across a supplication which reads, “O Fatima! What is preventing you from obeying…” However, we have come across another hadeeth narrated by Anas ibn Maalik, may Allah be pleased with him, which says that the Prophet said to Faatimah, “What prevents you from listening to what I am going to advise you, or to say when you wake up in the morning and in the evening, “Ya Hayyu ya Qayyoom, birahmatika astagheeth, aslih lee sha’nee kullahu, wa la takilnee ila nafsee tarfata ‘Ayn.” (O Ever Living, O Self-Subsisting and Supporter of all, by Your Mercy I seek assistance, rectify for me all of my affairs and do not leave me to myself, even for the blink of an eye.) [An-Nasaa’i]
There is no mention of Al-Firdaws in this hadeeth.
Allah knows best.