Persistant cough in patients with dementia is often due to dysphagia, or dysfunctional swallowing. Alzheimers patients lose the ability to swallow as the disease progresses and nears the end. This leads to coughing and aspiration of food or fluids. Offering small amounts of food or fluids at regular intervals helps. It sounds like it may be very appropriate to consider palliative care and hospice services at this point. I hope this helps.