Chronic Kidney Disease (CKD) exacerbation: Given the patient's history of hypertension and the presence of mild Acute Kidney Injury (AKI), a worsening of CKD is plausible. The mild polycythemia could be a response to decreased erythropoietin clearance or production, although this is less common. Fatigue is a common symptom in CKD due to anemia, electrolyte imbalances, or the disease process itself.

Infection: The very mild leucocytosis with a slight elevation in monocytes could indicate a low-grade infection, which is common in long-term care settings. Infections can cause fatigue and mild changes in blood counts.
Dehydration: Often seen in elderly patients, especially those with dementia who may have impaired fluid intake. Dehydration can exacerbate AKI and cause fatigue.
Medication Side Effects: Given the patient's history of glaucoma and hypertension, they are likely on multiple medications. Some of these could contribute to fatigue or affect kidney function.

Sepsis: Although the white blood cell count is only mildly elevated, sepsis can present subtly in the elderly. It is crucial to consider sepsis due to its high mortality rate if missed.
Pulmonary Embolism (PE): Polycythemia can be a risk factor for thrombosis. A PE could cause fatigue and mild changes in blood counts. It's a critical diagnosis not to miss due to its potential for high mortality.
Sleep Apnea: Common in the elderly, especially those with vascular risk factors. It can cause significant fatigue and could potentially contribute to polycythemia due to hypoxia-induced erythropoietin production.

Polycythemia Vera: A myeloproliferative disorder that could explain the polycythemia but is less likely given the patient's age and the mild nature of the polycythemia.
Erythropoietin-producing tumor: Rare tumors can produce erythropoietin, leading to polycythemia. This would be an unusual cause of the patient's symptoms but should be considered if other explanations are ruled out.
Chronic mountain sickness: If the patient has recently traveled to high altitudes, this could be a consideration for polycythemia, although it's highly unlikely given the context of long-term care.