Is it a condition for the wife to enter the patient's room to receive the reward for visiting, or is it sufficient for her to be present with her husband and inform the patient of her presence?

The Shari'ah encourages visiting the sick, and this is one of the rights of a Muslim upon his Muslim brother, with no distinction between men and women in this regard. There is nothing to prevent a wife from visiting the sick with her husband, provided that Islamic guidelines are adhered to, such as avoiding seclusion (khulwa), wearing the prescribed hijab, lowering the gaze, and not speaking in a soft and alluring manner. The wife has received the reward for visiting the sick, as long as the reason for her not entering was obedience to her husband.