I'm sure there is a lot more to know about the particular circumstances of this case, especially what led to this mother and babies leaving the women's shelter and the police being left to find them a roof over their heads. Time after time police are left to deal with people who have mental health problems (as I think will be confirmed eventually, IMO).
I don't think police get enough training in this area but I would have hoped workers in a women's shelter would have some. It still isn't always easy to know when someone is just being "difficult", or if they are ill or at risk to themselves or others.
With the twins at just ten months old the first thing I would think of in this case is severe post-natal depression. I used to work occasionally in the mother and baby ward of a private psychiatric hospital (UK). I think nearly all the cases I saw were NHS funded as these specialist units are few and far between. We could have the mother and baby together but with close 24 hour supervision by trained staff so they hopefully could maintain their bond but the staff were there to look after the baby if the mother was overwhelmed/unwell. Most patients were voluntary but we did have the ability to request to section patients under the Mental Health Act for short/long periods if they were a risk to themselves or others. It could take many months sometimes to get the medication and support working properly so these mothers and babies could go home safely.
It's just a tragedy when people fall through the cracks like this, it would be a lot for any woman to cope with, twins on her own and homeless. But for some women and their babies the hormonal imbalances caused by pregnancy (especially twins), followed by a lack of support can be catastrophic.
This case immediately on the surface reminds me (as it will others) so much of Andrea Yates who tragically also drowned her five children in the bath. If there was ever a textbook case of failures in mental health treatment and dire consequences of too many pregnancies it's hers.
I wonder if this mother asked for help from people around her as well as on Facebook, or if she felt she couldn't admit "failure" as she says. So sad she couldn't connect with someone who could give her practical, hands-on support.