Is it permissible to say "Allah is sufficient for us, and how excellent an agent is He" only after exerting effort to solve a problem?

This statement is incorrect. Rather, it is permissible for a Muslim to recite the prescribed dhikr (remembrances) in every situation. It is authentically reported from the Prophet, peace be upon him, that he said, "Allah is sufficient for us, and [He is] the best Disposer of affairs," during the expedition of Hamra' al-Asad. Regarding this, Allah Almighty revealed: {Those to whom people said, "Indeed, the people have gathered against you, so fear them." But it [merely] increased them in faith, and they said, "Allah is sufficient for us, and [He is] the best Disposer of affairs"} [Al 'Imran: 173].

It is stated in Sahih Al-Bukhari that Ibrahim (Abraham), peace be upon him, said it when he was thrown into the fire, and Muhammad, peace be upon him, said it when he was told: "Indeed, the people have gathered against you, so fear them." But it [merely] increased them in faith, and they said, "Allah is sufficient for us, and [He is] the best Disposer of affairs."

The hasbalah (recitation of "Hasbunallahu wa ni'mal wakeel") is established in the morning and evening adhkar by saying: "Allah is sufficient for me; there is no deity except Him. On Him I have relied, and He is the Lord of the Great Throne" seven times. Whoever says it, Allah will suffice him for whatever worries him in the matters of this world and the Hereafter.