Farringdon has great travel links - you can get their from nearly anywhere in greater London - all depends on how much of a commute you want. Depends how close in you want to be for the other activities in life.
I work in Farringdon and people in my office come in from Barnet, Harrow, Rayners Lane, Brixton, Herne Hill, Islington, Highbury, Teddington, Blackhealth, St Albans, Clapham, Brighton, Chislehurst, Wandsworth, Finsbury Park and somewhere in Cambridgeshire - to just name a few. Only the chap In Cambridgeshire has school age kids - but commutes for over an hour each way.
I think one of the property sites maybe Rightmove can allow you to do a search by commute distance now.
London is huge but also agree with a 10 year old I would be thinking more about secondary than primary.
Offer more info and maybe we can help - so what kind of budget etc if money is no object and you can afford private I would look at Islington if money is tight I would look at good commuting places in the South east which are on the Thames Link.