I don't think you are a tw*t at all. I think being a mother is an incredibly thankless task and has little immediate reward and it encourages the need for external approval. Getting a first is external approval of something you are doing and I understand that need completely.
I am doing an MSc at the moment. I started it when DS was 1 and DD 5 and I thought, with those pressures (DS at home with me full time) I would be luck to pass. I did my best for the first module at got 2% off a distinction so I thought, darned if I'm going to settle for a pass with grades like that, I'll push myself for a distinction (MSc level have just a pass and a distinction, no middle ground and I knew I was better academically than people who were just scraping passes). Since then I have had a good distinction in every module, despite having power of atorney over my elderly aunt, a father who was given 2 weeks to live 10 months ago, 2 small children, one of whom is home with me full time and also 2 part-time jobs in the evenings and weekends.
For me, it has paid off. I realised that to do what I want to do I need a PhD and viewed the two years of my MSc as a 2 year job interview, I did my best with everything, talked with and got to know the lecturers etc, and when it came to putting in my application for ESRC funding for the PhD, the staff were all very keen to help me improve it, because they knew me as a hard worker. I got the funding and I start the PhD as soon as this MSc is finished (I have one week off for a holiday in the sun!).
Wafle, waffle, waffle, but what I am trying to say is. Don't beat yourslf up over your determination to do well, it is understandable and that kind of determination will help you if you decide to develop a career after studying. However, don't beat yourself up over every grade and every assignment. Do what you can.
Time-wise. Study every evening and just take it easy with the kids during the day. Delegate what you can to your partner/friends etc. Let the housework go a bit but make sure you've done any evening jobs before the kids go to bed so you can come straight down and study. You'll find you actually get a lot more done with much less time because you'll be focussed.
I am getting to the end of my MSc now, all modules are done, I just need to finish my dissertation now by mid September. Now that I have the PhD funding, people wonder why I just don't think ah wel;l, it's just an MSc and ease off the study for the disertation and get an easy pass overall. I can't do that. I have worked so much and like you say, sweated blood for so long that I'm not giving up now - I will get a distinction to make all that worthwhile. I got a 2:1 in my first degree 12 years ago and always regret that I didn't get just 2% more and a first overall. This time I'll get it.
With regards to your kids, it is actually a very good role model for them to see their mother working for something and dedicated to a cause.
Good luck with it all and remember it is manageable, just be determined and organised and you'll do it.