Yes, your sister in law is being outrageous. Unacceptable.
First, sit down with your husband, and decide what the options are, and what you yourselves actually want.
For example, could you cope if you had a 'share the care' of your MIL, whereby she split her time equally between you and your SIL? (Are there any other children around, or is it just your husband and his sister?).
That might be option one.
However, if you feel that you actually couldn't cope AT ALL any longer, even with six months of the year (say, every other month or whatever swapping your MIL between yourselves and her daughter), then not even that solution would work. In which case the only option really for your MIL is residential care in a care home.
Of course, your sister in law might decide (as seems likely from what you say) to refuse to have ANY care of her mum, and simply want her put in a residential home anyway if muggins (that's you!) is finally starting to object to her outrageous intent of dumping all her mum's care on YOU!
A third option might be for you to have your MIL (leaving SIL out of it), but 'sharing the care' not with SIL but with a LOT of residential respite care for your MIL (say, every other month or whatever you feel you can cope with.) This might be a little less expensive than full time residential care, but you might have difficulty in finding that much residential respite care available?
One absolutely essential thing to do, however, IF you end up doing ANY of the care, but your SIL does ZILCH, is to PAY YOURSELVES for whatever amount of care that you do. Otherwise you are simple 'subsidising' your exploitative SIL.
As others are pointing out, of course, if your MIL has sufficient funds (property and savings worth more than £23k, the current threshold set) to mean she has to self-fund any care, including respite residential and full time residential, then your SIL might suddenly realise that 'putting mum in a home' will wipe out any hopes of her inheriting anything when her mother dies - the money will have gone on the care home! (Care homes will cost at least £100 a DAY - that's what my MIL's costs!)
Many families 'drift into caring' without really realising the implications, and that it is all too often a 'lobster pot' - the caree seldom 'recovers' sufficiently to go back to living independently. And even if they do for a while, old age keeps on going remorselessly. Your MIL already sounds in a pretty bad way (incontinence is always a bad sign, alas), and will simply get worse and worse.
She may have already passed the point where anything other than full time residential care is possible.
Whatever happens, the current situation can't continue. Personally, I would insist on a 'back payment' from your MIL to cover the cost of what you have done already, to stop that amount of money being 'filched' by lazy SIL when her mum does die!
(Oh, and by the way, any 'whining' by SIL that she 'cannot' look after her mum because of blah blah blah whatever, but you CAN, is utterly irrelevant. The ONLY point is 'what is fair' - and what is NOT fair is the current situation, and that's that.)
