How can I help my friend, who is suffering from a poor psychological state due to her father's illness, to strengthen her trust and hope in Allah Almighty?

All of Allah's decrees for the believer are good. What afflicts her father in terms of illness is a mercy from Allah. If he shows patience and seeks his reward from Allah, he will have immense recompense. The Muslim should remember Allah's vast mercy and His blessings, and that Allah is more merciful to His servant than a mother is to her child. He is All-Wise, placing things in their rightful places for a great wisdom for which He deserves all praise.