My read is that this lady lives by playing on the sympathy of others. The 2 or 3 days may be weeks or months. She is no man’s dream of a perfect relationship. My suggestion is you give the bf a time limit on his generosity. Tell him you will be seeking greener pastures if the limit is exceeded, and act accordingly.
Are 'her children' also his children? If so, you don't have much ground to stand on if she needs help housing them. Even without that, kids without housing is a tragedy and I'd try to drum up some sympathy. If she is his ex, she is his ex. Do you wonder if she is really his ex or he's sneaking around? That can happen if she lives there or not.