To be fair to the lefties, I would respectfully suggest that the figure of £11,000, which I presume relates to day fees only, is suspiciously low. Three terms at a run of the mill (ie not a big name school) fee-paying pre-prep (Infants) will cost you £10,000. "Name" pre-preps could be asking £15,000 for three terms.
Prep fees are higher. Senior school fees are higher again, with the most expensive in the country costing more than £20,000, and that's ignoring the boarding fees.
For the average to only be £11,000 there would need to be far more pupils in private pre-prep than there are in private senior. Something like a six or seven to one ratio. I'd treat that claim of £11,000 as the average with a fair degree of scepticism.
Swings and roundabouts though isn't it? I'm sure many state schools in London spend more than 10k per pupil.