Of course it is illegal - it is defrauding the benefits system.
You would all be breaking the law.
Firstly, incapacity benefit is not means tested so the mother will not lose that benefit.
Council tax - there is an automatic 25% discount for those living alone. I'm sure your sister could pay this - probably around �250 a year as an average.
However, incapacity benefit is not enough to live on, so unless she gets DLA (another non-means tested benefit) she will be on some form of income support which is means tested.