This might be a long shot... but how does she interact with your children? Is she warm/happy/comforting/loving/calm with them?
It may be possible that she has been suffering from
postnatal depresssion (which I think Maricha75 was referring to?). Her lack of awareness over your kids, extreme sensitivity emotionally and in particular the insomnia/trouble sleeping are all key 'flags' of postnatal depression.
I don't want you to be alarmed or think I'm jumping the gun or anything, but I do suggest you consult a doctor about your wife's behaviour. If you want, there is a self-assessment 'test' for depression here:
Postnatal Depression - Symptoms - NHS Choices
But a professional is obviously better...
In the mean time, make sure you child proof your house thoroughly.
When was the last time you and your wife had a day's break? Is there a family member/close friend you can leave your kids with for the day? Perhaps a relaxing 'day off' will calm your wife long enough to open up a bit and then you can talk to her more deeply without worrying about where the kids are climbing.
I wish you both the best... and as alarming as it sounds, please do check out postnatal depression as a possibility; sometimes these things do just 'happen' but if it's left untreated, it can become a long-term problem.