Hi,The lottery is a silly fantasy and a complete waste of money. My mother is the product of her own ruined childhood. All I would wish is to have a place where I could have her with me (and a carer), with garden and dogs. Keeping her at arm’s length, but basically cared for NOT in an institution.
In my view, I don't think it realistic to hope for a "place" in which anybody is cared for that is not an institution, other than i) their own home - which costs - or ii) someone else's home - if there is someone else to care, which you have already denied - or iii) a hospital if and when required or iv) a care/nursing home - which costs - or v) a hospice - for end of life care.
You have already said your mother is highly intelligent and living at home with dementia. Does she still have the mental capacity to manage her own affairs? If so, then she can decide what, if any, help she wants to accept. If not, then either her PoA has to decide or the medics do.
Either way, as time goes on dementia only worsens.
In earlier posts you state you cannot live with your mother, but in the latest post you state you want her with you, but at arm's length.
I wish you well.