@Howdoyousolveaproblemlikemil Sorry if you've already mentioned this, but does your MIL have any income at all besides the money she gets from her ex husband? No pension? Is she entitled to any benefits?
I think it's pretty scary that she had £180,000 from the sale of her previous home, gave away almost a quarter of it, and now only has £100,000 in equity in YOUR home, plus £40,000 in savings assuming she hasn't already spent that.
The problem is that if you manage to achieve your preferred option, which would be buying her out, she's then effectively homeless with £100,000-£140,000 in the bank. If that's not enough to buy even a one bedroom flat, she won't be able to raise a mortgage for the rest. Does that mean she'd have to run down her savings on rent until she was eligible for benefits? If she's still in her 60s that money will run out frighteningly quickly and then she will be on the bones of her arse.
But then what happens if she carries on living with you indefinitely and eventually needs to go into a home? Could you and your husband be forced to sell your home to pay for her care fees, or alternatively pay her care fees up to 10% of the value of your home in order to avoid selling? Because in that scenario, not only is there zero inheritance (whilst your BIL and SIL have already had £20,000, thank you very much), you and your husband could find yourselves entirely on the hook for her care fees whilst BIL and SIL can completely wash their hands of the whole thing.
Was this aspect discussed at all when you decided to do this?