Sleep deprivation is among the most common complaints about the ICU experience. ICU sleep tends to be light and non-restorative (as opposed to deep/restorative sleep), severely fragmented, and distributed throughout the day and night rather than consolidated into nighttime hours. Sleep-deprived patients suffer from sleep debt, a condition of impaired attention and memory, and cognitive slowing. Sleep disturbances in the ICU arise not only from light and noise pollution, but also from drugs that interfere with brain activity involved in restorative sleep. Sleep deprivation has also been suggested as a major modifiable risk factor for acute encephalopathy, also known as delirium. Delirium is an acute state of confusion that affects up to 50% of non-ventilated ICU patients and is one of six leading causes of preventable morbidity and mortality in hospitalized elderly patients. Many patients who survive delirium experience long-term cognitive impairment and loss of independence. Current medications used in the ICU to treat sleep problems (e.g. benzodiazepines, antipsychotics) do not induce natural sleep and do not prevent delirium. In contrast, the investigators have found that the α2-adrenoceptor agonist dexmedetomidine can induce biomimetic sleep, a brain state whose pattern of electroencephalogram (EEG) activity, cerebral blood flow, and functional connectivity approximates restorative sleep. Moreover, a recent large clinical trial in post-surgical patients suggests that low-dose dexmedetomidine given overnight substantially reduces the risk of delirium. It is unknown whether this benefit is linked to improved sleep, or whether patients with better sleep while in the ICU have better long-term cognitive outcomes. The investigator's central hypothesis is that sleep deprivation substantially mediates both the short- and long-term cognitive impairments associated with delirium in critical illness. To test this hypothesis, the ICU-SLEEP trial aims to: 1A) Compare the burden of delirium, as measured by the number of delirium-free days (DFDs), in ICU patients non-ventilated at study enrollment, who are receiving biomimetic sleep induced by Dex, given as a continuous overnight 1) very-low-dose or 2) low-dose infusion vs. 3) usual care and placebo; 1B) Assess whether the 1) very-low-dose continuous overnight infusion of Dex increases DFDs compared to the 2) low-dose continuous overnight infusion; 2A) Determine whether Dex reduces ICU delirium via reducing sleep deprivation, using causal mediation analysis; 2B) Determine the associations between specific components of acute cognitive impairment, seen in sleep deprivation and delirium, with specific measures of sleep deprivation; 3A) Determine whether ICU patients treated with Dex while in the hospital have a lower incidence of long-term cognitive impairment; 3B) Determine whether any differences in long-term cognitive impairment between ICU survivors treated with Dex vs. usual care and placebo are mediated by differences in sleep deprivation. The knowledge gained will provide insights into the mechanisms by which Dex prevents delirium and guidance on the optimal dosing strategy for prophylactic Dex, which can be used to design pivotal phase 3 clinical trials.