There is a lot of confusion regarding “last rites” and anointing of the sick.
First, the sacrament of anointing the sick. This is a sacrament which a priest or bishop can perform on someone who is gravely ill. There is, of course, a prescribed rite to be followed and the Oil of the Sick is consecrated at the Chrism Mass of the diocese each year. A good priest will always carry a small amount of this oil on his person in an “emergency kit”.
Now what was, and is still called “last rites” or “extreme unction” is not a single sacrament. It is a collection of sacraments and prayers meant to be used with people on the verge of death. This is the important part, anyone who is sick can receive the anointing, but you must be close indeed to death to receive last rites. Its major parts are three sacraments of penance, the Eucharist (called Viaticum), and anointing of the sick. Naturally, only a priest or bishop may perform penance and anointing, so if only a deacon or a lay person is available, the dying person may be given Viaticum and attempt perfect contrition for his or her sins.
When there is danger of death, even a suspended or laicized priest may offer these sacraments to the dying. It is the primary instance where the care of souls overrides the ordinary discipline of the Church.
So in relation to the OP’s question, a person with Alzheimer’s can certainly receive anointing of the sick, and it can be repeated as your priest sees necessary. The patient can also regularly receive ordinary sacraments of penance and the Eucharist, as ministers are available to visit. An EMHC would be able to share Scripture and pray with the patient in the course of his or her visit. In the event that the patient is on the brink of death, then you may certainly request a priest to perform last rites.