Final answer:
The hormone likely malfunctioning in a patient producing over 10 liters of dilute urine per day is the antidiuretic hormone (ADH). ADH is crucial for the reabsorption of water in the kidneys, which concentrates the urine.
Step-by-step explanation:
A patient in the ICU producing over 10 liters of dilute urine per day is likely experiencing inadequate functioning of the hormone antidiuretic hormone (ADH). Normally, ADH works by promoting the insertion of water channels, also known as aquaporins, into the cell membranes of the collecting ducts in the kidneys. When ADH is functioning properly, these water channels allow for more water to be reabsorbed back into the bloodstream, thus concentrating the urine and reducing its volume.