@WelliBob I think you are going to struggle if you want solely day and also solely coed and that's because the good single sex day only schools are so good they haven't had to go coed, and also the good boarding schools are so good they haven't had to go solely day.
The thing about boarding schools is that they are often really amazing for day pupils. I get what you say about being set-up for day being a strong preference, but in a traditional public boarding school, generally the staff are paid more and possibly given housing, even without working in be boarding side, so it attracts a very high Calibre of academic staff, with a lot of competition for staff places.
There are also many more endowments there because of the history and age of the school (and because they started as boys' schools, all girl school are so poor in comparison) which generally leads to more money, more facilities, more scholarships available and more opportunities.
For activities, again it's set up to offer more and staff give more of their time, for example traditionally coaching sports teams alongside PE staff so all the teams end up being strong or offering clubs and extras outside the timetable (I have yet to see things like rocketry and astronomy clubs offered in day schools, although I am sure there must be some somewhere, but again the only contenders are most likely to be in London!!)
Maybe your best bet is to go through the good school guide/Tatler/HMC and mark out some on a shortlist?
But the main reason I prefer boarding schools to day is it's also a massively friendly place to work and I go in smiling every day. I think that's because it feels like a community in a way that a day school just doesn't.