I really identify with your money worries and renting a place you can ill afford - I have spreadsheets for spreadsheets now! I know its hard, its stressful and causes you sleepless nights but something always does seem to come along just at the right moment; just keep everyone in the loop, explain the situation and most people/companies are open to helping where they can.
I agree with job, please don't even consider this new place, it sounds like it should be condemned, on top of that you would not be happy there which will affect both you and your d emotional and physical well being, sometimes no matter how hard it gets we have to be rational and draw a line at what measures we are prepared to take - living in a "dump" should not be one of them. Dry, clean and safe - you deserve no less.
Its such a sad situation with this little girl and I can see it is tugging your heartstrings, it would mine too - but I have learnt valuable lessons over the years by helping families who just wont help themselves and it always ends the same way - them doing the take take take. Its a tough one for sure; as long as you are confident she is getting her basic needs met and she is being cared for then I would not encourage her calling on you - but I have to say that if I felt her and her siblings are not being cared for properly or she could be in danger then I would involve the authorities - its the responsible thing to do.
I think you are doing great, you are still moving forward and dealing with the challenges that come along with grace and strength, you may not see it, but I do. Keep going Heather, you will get through this crappola time and life will be good once again.