The honest answer - IME and IMO a City firm will not be able to offer you the type of work life balance that you are seeking. There is a reason why the money is good.
If you charge clients a high hourly rate, you can pay staff more. However, people will only pay higher fees if there is a benefit to them. Sure, part of it is about technical expertise (particuarly in the techy areas like tax) and sometimes it's about brand name, but for the most part what they are buying is a responsiveness and turnaround time that fits with their needs. That means that, if you need to stay until 10pm to deal with their transaction, that is what they expect rather than a day's slippage in the timetable. If they call you at 5pm "I'll do that tomorrow" is still often not fast enough. As they see it, that is why they are paying the premium.
What that in turn means is that flexible working is very hard to accomadate. If you've left the office for the day, or it's a day you don't work, your colleagues X and Y have to do the work instead. It doesn't wait for you to come back.
Law firms in the City get a really bad rap for being family unfriendly. But, TBH, I think it's a question of 'you pays your money, you takes your choice'. A lot of them would love to be more family friendly, but know that the only way to do it is to dump work onto others, which is appalling for morale. If you want to be flexible in a City firm, it's not going to be a case of getting set hours 9-5. It's more likely to be working from a home office one day a week or a committment that you can leave the office at 5 if you log back on from home once the kids are in bed if needed.
The other thing I would say is that you should beware of what people offer. I know someone who moved firms on the strength of family friendly promises. I am sure that the firm meant them at the time, but six months down the line most of it had slid by the wayside.