From what the OP said, **it would seem that the mother is defrauding the state - she said that her mentally ill sibling cannot be seen to have money in a bank account lest the state benefits stop coming. This is illegal and immoral, if it is happening. The asset of the house, if left to the mentally ill sibling, would be taken by the state to compensate for payments already received, and rightly so. It is not the taxpayers’ responsibility to pay for care of someone if there are assets that are being hidden.
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My advice is for the OP to seek spiritual help from her priest and get legal help to deal with this sticky situation. She is not obligated to care for her mentally ill and destructive sibling, although making sure the sibling has some sort of help would be charitable.
My suspicion is that the mother has enabled a lot of bad behavior and if the sibling had had to be more accountable, at least some recovery may have been possible. I may be wrong, but I have seen this before. The mother (or parents) protect the mentally ill person and actually make things much WORSE. Once the parents are out of the picture, sometimes recovery is possible, but many times, it’s too late.