It sounds as if you already have some proof that it is your home and you are contributing towards it, even if not paying actual rent. Try to persuade your gran to change the bills into your name - just to make it easier and above board. Until then keep all your bank statements to prove the payments come from you.
Going into care considered nowadays as the last resort - most have to fight to achieve it. However, you should get advise from CAB etc re your rights and how to get care. Also do keep trying to get her to see her doctor. Memory loss and confusion can be a symptom of other illnesses such as, I believe, bladder infection. It could be something simple and curable.
Coming soon! Spot a field, spot a cow.......

