Having suffered postnatal depression (PND), its just possible that this caused her to think suicide was the answer. I'm not sure though how people can suggest her family failed her. If she had PND it wasn't their fault. They had no idea she was going out to kill herself. How can you blame them?
PND feels bad because having just had a baby you are supposed to be at your happiest and most joyous - isn't that what bringing a new life into the world is all about? Truth is not enough is known about PND. Yes its probably hormonal but its also more likely to happen if there has been previous incidence of depression. And depression I am afraid still carries a very high stigma. So being depressed and/or having PND may have lead to this woman deciding she could not continue. She may have felt great shame at the diagnosis. What would people say? She should be happy! She may have also been put onto anti-depressants that day and told she could not breastfeed her child because of them. This would be another blow - the guilt and shame of being too ill to feed her child in a way that modern women are constantly pressured to do. Breast may be best but making women feel guilty if they cannot manage it is sh*t.
She may also have had finance worries (which are inevitable when another child arrives [unless you're comfortably well-off]). Her relationship may have been in trouble - just because she'd just had a baby doesn't mean she was happy in her relationship.
Having had Ante-Natal Depression and Post-Natal Depression and been treated for the latter with anti-depressants, and had thoughts of walking out in front of cars, taking pills etc, I can only feel sorry for her family as they probably had no idea that she had such painful thoughts and feelings about her life.
Rant over. Does that make it more understandable why she dowsed herself?
http://en.wikipedia.org/wiki/Post_natal_depression