To what extent is the hadith authentic in which Aisha narrates her astonishment at the Prophet’s prolonged standing with Dihyah al-Kalbi, his permission for him to enter at any time, and his informing her that he was Jibreel (peace be upon him)?

It was not mentioned in this context, but its origin was narrated in some Sunnah collections. Al-Nasa'i narrated from the Prophet, peace and blessings be upon him: "And indeed, it is Gabriel, peace be upon him, who descended in the form of Dihyah al-Kalbi," and Al-Albani graded it authentic. Al-Tabarani also narrated from Anas that the Prophet, peace and blessings be upon him, used to say: "Gabriel, peace be upon him, comes to me in the form of Dihyah al-Kalbi." Similarly, Ahmad narrated from Ibn Umar: "And Gabriel, peace be upon him, used to come to the Prophet, peace and blessings be upon him, in the form of Dihyah," and Shu'ayb al-Arna'ut graded its chain of narration authentic according to Muslim's conditions.