How to remain tactful when you want to scream "You live in squalor, I'll never send my daughter to your home!"?

I think you just avoid the issue of her dirty home and get in touch saying a relative has offered to look after your daughter for free, you've managed to get her into a nursery/daycare you were on the waiting list for, or you've found someone to look after her closer to your house that would work out better for you. The list of excuses is endless, just pick one and make the call! How disgusting though! If you want to be honest with this woman then don't beat around the bush, take a deep breath and tell her that you feel her house is dirty and therefore an unsafe environment for your little girl - good luck if you go with this option!