Hi, let's just have a look at para 317 of the Immigration Rules :-
317. The requirements to be met by a person seeking indefinite leave to enter or remain in the United Kingdom as the parent, grandparent or other dependent relative of a person present and settled in the United Kingdom are that the person:
(i) is related to a person present and settled in the United Kingdom in one of the following ways:
(a) mother or grandmother who is a widow aged 65 years or over;
.....(other possibilities not relevant here have been deleted) ............
and
(ii) is joining or accompanying a person who is present and settled in the United Kingdom or who is on the same occasion being admitted for settlement; and
(iii) is financially wholly or mainly dependent on the relative present and settled in the United Kingdom; and
(iv) can, and will, be accommodated adequately, together with any dependants, without recourse to public funds, in accommodation which the sponsor owns or occupies exclusively; and
(iva) can, and will, be maintained adequately, together with any dependants, without recourse to public funds; and
(v) has no other close relatives in his own country to whom he could turn for financial support; and
(vi) if seeking leave to enter, holds a valid United Kingdom entry clearance for entry in this capacity.
Notice that each of those is joined by an "and". So it seems that is it suggested you are falling down on :-
(v) has no other close relatives in his own country to whom he could turn for financial support
But, based upon what you post, she does indeed have a close relative ... your sister. I recognise that this might appear to be a blunt question but .... why can't your mother move in with your sister?
Also (iii) is clearly a problem given, as you say "My mother is not hard up financially".
Unless you prove that it totally impossible I don't see how your mother can get a visa.
The visitor visa? That might be problematic. After all the ECO might think that your mother would not return at the end of the visa period. And you yourself have said that she cannot now live alone.
Sorry if this all sounds a bit harsh. But it is how I see the situation. Just because you are British does not give an automatic right for parent(s) to move to the UK.