I think you're well-intentioned and trying to do a good thing. But if you're going to do that good thing, you DO have to initiate contact with the child's mother. And probably her father. And you need to consider the other two children in this situation. This good thing you're trying to do is a really big project. If you're going to take it on, you need to go above and beyond the efforts put forth by the other adults involved in the situation. You need to call the kid's mother, fill her in on what her dd has been doing, ask how mom is doing, and make a plan for how long you are keeping her dd. You probably need to contact her dad. You need to arrange regular contact for communication about the child, and you need to follow through on that communication even when the kid's mom and dad seem willing to let it drop. If this child stays with you in a long-term, formalized way, you will need to facilitate visits with both parents and with her siblings.
A social worker looking at the situation will certainly agree that mom's failure to contact you means that she doesn't give a crap. They may see your failure to contact mom as evidence that you don't give a crap either. That's not what a caring adult would usually do when trying to help a child. They may also be concerned that you didn't contact CAS when Mom left the country and left the kids without food. That situation would have compelled CAS to act, but you didn't report it while it was happening. That's a pretty big missed opportunity to get services for the children.
It sounds like you're a caring and generous person. It sounds like you feel exploited by this mother. If you're going to bring this child into your family, you will not just be helping her. You will be creating a relationship with her entire family, and she will be bringing her own contributions in to your family. If you understand what that will mean for your family (and it will not be like having her as a house guest for a month - being part of a household is not like being a guest in it) and you are ready to build that relationship and accept those contributions, then you can probably make this work.