Hi Mariagm
I am in the same position as you, at the same stage. There are loads of posts about the problems of getting the LA to agree to residential care. You can check out mine here
http://forum.alzheimers.org.uk/showthread.php?81574-I-feel-so-guilty-but-Ive-had-enough
It's a long thread, you might want to scan it quickly, but there is some excellent advice in it from other TP'ers about what to do.
My LA are still talking about keeping Dad in his home even if I refuse to care for him anymore due to my own stress, ill health, full time job and burnout (he is alone, my mum died last year) - good luck to them. The LA is millions of pounds in debt so they want as few people in care homes as possible because it costs £1000+ just for minimum care in a care home (people with more demanding needs have to pay much more).
As one TP'er said to me, you might just have to let them take the strain by refusing to care for him anymore. Basically I think both you and your Mum will have to refuse before they will carry out an assessment of his needs with a view to placement. If he is judged incapable after an assessment they will then put a care package in place, which may involve a move to a care home or care in his own home provided by outsiders.
It is absolutely heartbreaking to refuse to care. I don't want to do it but I have to for the sake of my health. If your Mum wasn't around, what then? They would have to care for him. You have a right to your own life, you are not legally responsible for your Dad.
Good luck - sending you hugs and lots of virtual support
Jane xx