American Indian Elderly and Disabled Access to Health Care Act of 2005 - Amends the Indian Health Care Improvement Act to allow the Indian Health Service, an Indian tribe, a tribal organization, or an urban Indian organization to pay the monthly premium of an Indian who is a Medicare part D (Voluntary Prescription Drug Benefit Program) eligible individual enrolled in a prescription drug plan or a Medicare Advantage-Prescription Drug (MA-PD) plan under Medicare part D.