It's good that your MIL is part way through a reablement program. You need to make sure that SS know about every incident and problem over the remaining time. If they've already lost the initial carers, it's looking like it's not going well.
We've been in a similar position with 2 relatives, at the same time, at opposite ends of the country. I also had young children - it was a tough time.
My advice would be to find a care home which you and DH like the look of, prioritising what your MIL would want, that has spaces free. Then strongly suggest to your MIL that she goes there for a while, just until she gets back on her feet. Once she is settled, she might well decide to stay. My Gran was relieved, although she wouldn't admit it, when she no longer had to worry about getting to the toilet, and also had her bills, cooking and cleaning all sorted. My other relative was not so happy in a care home, he went back and forwards between care and his own flat, but 1 out of 2 was massive progress for us.
I had power of attorney for both relatives - one granted before they were ill, another done afterwards, as he had capacity. This meant that I could deal with all the paperwork and practicalities of assessments, selling their property, paying bills in the meantime. This helped persuade my Gran, as she was worried about sorting out her retirement flat, which I could reassure her I was dealing with.
Good luck, I hope your MIL comes to realise she needs full time care of her own accord, it's early days yet.