Hostage in own house - farcical situation

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It's obvious SS don't want to take her into care etc, as they will have to pay, since she is below the threshold. That said, there is the pernicious issue of 'choice' etc etc. If she is not yet so far advanced in her dementia that she counts as having legal capacity then yes, they can't force her to go into a care home, or respite etc.

That really leaves the onus on you.

You have two choices. One would be, simpyly, to evict her. She's living in your home, it's not hers, she has no legal right to be there, so you could evict her to a care home....that said, the HOW of actually getting her physically out of the house is 'tricky' to put it mildly! How DOES one forcibly remove a 'lodger' who does not leave the house (does she?). Presumably it can be done, but it won't be 'pleasant' as she clearly doesn't want to leave, and whilst in a way I don't blame her (who DOES want to go into a care home?), on the other hand of course you and your wife want your life - and your house! - back to yourselves.

The second option is to modify your house. You say she's in a kind of annexe, but the key factor here is 'separate entrance'. If you can create for her quarters a separate entrance, and then a lockable door between you and her, then it becomes possible for you simply to tell her doctor and the SS 'We're not proving ANY care any more', and then they will HAVE to send in paid careworkers to take over.

You will probably find that easier than your wife, why may well feel guilty, etc, but really, all she/you has to do is what I call 'keeping company' - ie, you can spend time with your MIL, even have her to 'your side' sometimes, eg, for an evening meal, etc, but basically she 'lives next door' in the anexe, whgich is the only section the careworker accesses.

You don't have a legal duty of care, and you can wash your hands of her care, leaving it entirely up to SS to sort out.

All that said, to be brutal, if she is carrying faeces around in a handbag, that is pretty clear evidence that she is NOT 'in her right mind' in a seriously big way - so she might not actually have legal capacity any longer. Please make notes of all that she does that indicates dementia, and report them to her GP (in writing), and at some point she will be deemed to have lost capacity, and then she gets no say in what happens to her.

It's all very sad, very depressing and very frustrating - until my now 93 y.o MIL went into residential care it was a nightmare for me! It still is intensely sad to see her appalling decline.....of which she, blessedly, has no realisation at all, thank goodness. She'd loathe to know how bad she has got...
I can't see that carry faeces around in her handbag is "mild" dementia at all!
Who did the diagnosis? Is the CPN - Community Psychiatric Nurse - involved. If not, ask for a referral.
I agree with Jenny, make a separate entrance and lock the adjoining door.
I actually sleep in my own Granny Annexe, as after a car accident I could only crawl upstairs on all fours. I had two choices, move from my lovely New Forest cottage where I've lived for 40 years, or have a downstairs bedroom. My son converted my garage to a lovely bedroom, but I was adamant that I didn't want to access it through the main house. We have a large conservatory across the back of the house, and I get to my room through there, so that if I need any carers in later years, they won't come through the house.
The bedroom must have a window or two, so all you do is turn a window into a glazed door by removing the area below the window, not need for any lintels, because they are already there. A builder should be able to complete the basic work in one day. First they will mark the brickwork to be removed, get a huge disc cutter, cut through the bricks, and then knock them out. Disc cutters can make a lot of dust though, so be sure to cover as much as possible with dust sheets.
Remember, it's your house, and you can do what you want with it.