@hassletassle - I have followed your story throughout.
You have tried too hard for too long now.
Its critical that you prioritise your own emotional and physical well-being which impacts your ability to parent your own tiny children.
You are being exploited and exhausted here. This lady can find an alternative sponsor or housing. Please pass this on directly to the council with immediate effect. They can arrange a hotel or new sponsor. This is unsustainable for you. She will be fine.
Feel proud of what you have done. It’s more than enough. It’s enough now for all of you.
As others have said don’t get emotionally blackmailed or FOG (fear, obligation, guilt) about “kicking her out” or “making her homeless” - these are inflammatory words. Tell yourself it is time for her to move on instantly. She has networks, work etc - you have provided more than enough of a foundation. Do not explain, defend or justify your decision to anyone.
Keep it brief, factual, firm and friendly.
”Dear Liaison Officer,
The sponsorship arrangement with xxx will end in 7 days. Please contact xxx to confirm how you will support her transition to alternative accommodation.
Thats all that needs to happen.
It seems that right now you and your family are suffering more than this lady and her child. Please prioritise your needs.
Be proud that you have done more than 99.9% of the population.