Thank you for asking the question and performing this great action. In his book Mourning in Halacha (pp. 29-46), Rabbi Chaim Binyamin Goldberg has a number of recommendations based on Jewish law which would apply to your visits.
- It is desirable to explain to the patient that illness in this world is a kindness which the Holy One bestowed for mankind. The patriarch Jacob prayed and requested that people should become sick so that they could give their final commands to their family and so that their children would have time to come and be with their parent
- It is desirable to speak with the patient about the need for giving instructions and putting his affairs in order, e.g., if he owes money, or if others owe him; if he has money or property deposited with others, or if he holding deposits for others
- It is desirable to speak with him in a positive way regarding how he wishes his property to be disposed and what he wishes to command regarding personal and family matters. It is very important to him to explain the great spiritual merit that a person receives when he gives tzedakah (charity) before parting from his belongings
- One must be extremely careful when speaking to the sick person about these matters, to make sure that one's words do not cause them suffering and worsen his condition
Regarding prayers, Jewish law prescribes saying viduy, a confession of sins, before departing this world, as a great merit to the soul. This is also seen as a powerful aid for rapid and complete recovery. Legal texts state one should tell the patient "Most of those who confessed did not die, and many who did not confess died. Many who are walking the streets recite the confession, and in reward for confessing you will live. Whoever confesses has a portion in the World to Come".
The viduy is said near death. See here for more details, translation and transliteration.
When a person who is ill prays, it is very effective. Therefore the sick person himself should try to pray from the depths of his heart. Recommended psalms are some or all of psalms 16, 23, 25, 51, 91, 102, 103, 121, 139, 142. Additional recitations (e.g., Pitum HaKetoret) can be found in Jewish prayer books if you have access to one. See pp. 34-40 and 45-46 of Mourning in Halacha for additional prayers including for the final moments.
See also here from Chabad for prayers for final moments (with Isaac Moses' very helpful prioritisation and guide to these prayers).