YANBU. You have spent years of your life dedicated to becoming a lawyer and then years enjoying your job and career. I assume you also did not plan giving up work when you started this journey all those years ago.
Your DH also has a career that he has studied / worked for and has made a recent choice to take on a new job where he feels unable to request part time / flexible working. To some extent, it is his choice to spend long hours at work to provide for his family to the level that he does and to pay for private school.
Giving up everything you have worked for to sit at home watching daytime TV looking after a puking, pooing, crying machine can look rather unattractive on paper!
This needs to be a family decision with neither party feeling beholden to the other - you had this baby together and there is no rule that states that it should be you who gives up work, if at all.
There are numerous options, you could both cut your hours but both keep your careers, in the long term this could result in an overall net benefit to your family income.
There are many female city lawyers who find themselves in your position so a job share could be a real possibility if you worked hard at finding a job share partner.
It sounds like you are not quite ready to make this decision. There are huge implications to your future career and how you develop as a professional and as a person. That's not to say it's the wrong decision, just that you need to take time to make this decision and talk it through with your husband.
He needs to understand the impact of this decision on your sense of self etc and also that him leaving you to look after DD from 8 to 8 leaves you with 12 hours of childcare to do on your own. Effectively a new job with no colleagues and no training or experience. He needs to realise how lucky he is that you are willing to give up your career to look after his daughter.
That aside, if you do go back to work, I would think about a nanny - it is a very good alternative for young children who would otherwise spend long hours in nursery. Having them cared for in their own home can make things easier.
I was a professional with a great job that I loved and earned more than 50% of our household income. When my first DC was born, I was angry that I had spent so much time on qualifications / career / lifestyle that left me unable to stay home and look after my baby but I was unable at that time to make the changes that would mean I could. I was also not ready to give up my career, my working life which defined me and who I was to become a SAHM, a role I knew little about.
My DH said at the time that he would consider being a SAHD but when push came to shove and to opportunity arose for him to do that, he didn't take that opportunity. 3 years later, our circumstances changed, I had changed and jumped at the chance to have some time out with my DC's. For now, my role as a Mum primarily defines me and I love it. At sometime in the future, I will work again and be defined by the sum of the parts.
It's not always straight forward but the most important thing is that you and your DH agree with each other and understand the others choices, motivations and feelings.
Good luck and sorry for long post.