A lot of private schools have a Year 9(13+) entry but the entrance exam can be quite challenging if your DC is coming from a state school.
The 11+ is something you can tutor for yourself but at 13+ the child will be examined on science and languages plus other 'core' subjects. Consequently the odds will be heavily stacked in favour of the highly tutored kid or the prep school kid.
The hidden costs? The schools usually have annual language trips to France, Spain and Germany at about £500 a piece. Out of 150 kids only about 30 go on any one trip and very few go on more than one. Then there are the history trips, the sport team touring trip, the ski trip and so on.
You can in theory spend about £3000 pa on school trips but in reality many parents just pick one or none at all. Out of 150 kids only about 30 kids go on a particular trip so its not like the child who doesn't go is made to feel like the poor relation.
Hidden costs aside, fees are currently between £9k and £14k. Reading other threads, some in the London area charge up to £18k with some up to £30k. Even assuming that its £12k a child average then its do-able with an income of £50k since DC2 is 8 years away from secondary school.
Whether I would recommend it is another matter :) The fees are only going to go up, usually ahead of income increases and inflation so if you are borderline now then you will have the anxiety of always having to keep one eye on household expenses.
But then, I suppose that once DC2 goes to school you'll be going back to work and by the time she is 11 you would hopefully have saved up enough for her.