£10k a year?? For secondary? None of the ones I've looked at or any that DH has ever worked at.
The figure I gave covers all independent schools, both primary and secondary. However, I can point you at two independent schools near me where the fees at secondary level are just over £9k. In general independent schools in the north are substantially cheaper than those in the south. Schools that do not offer boarding are cheaper for day (i.e. non-boarding) pupils than those that do.
Do private schools, especially the big, long established ones, have a lot of capital?
Some do, some don't. Some used to have large reserves but have seen them depleted due to low numbers of pupils in recent years. I don't have a source for a simple answer as to the typical level of reserves. However, capital spending by independent schools averages around £1,500 per pupil.
And how about the money that state schools get from central government for things like building work?
Varies enormously from school to school but adds around £1,000 per pupil to the figure I gave previously.
What about academies - how much difference does sponsor money make?
Again, varies enormously. Some sponsors don't provide any money. Others do. ARK, for example, adds roughly 10% to the amount received by the government.