Egglescliffe to London Kings Cross is operated by Grand Central and has 5 direct trains per day. Journey time is 02h 40m to 03h 03m.
Darlington to London Kings Cross is operated by Virgin Trains East Coast and has 31 direct trains per day. Journey time is 02h 23m to 03h 15m(Due to late night restrictions).
Plus there are many more indirect trains between the Tees Valley area and London which will get you into centre of London in under 3 hours.
London City might work but I'm not sure if the demand would be enough to support that route. Airlines must look at London's major transfer hubs Gatwick or Heathrow or forget about it. The likes of Southend / Luton / Stansted would be pointless. I would prefer Durham Tees Valley Airport to focus on overseas airports that can also be used as hub airport instead of the London hubs. i.e. Target Dublin, Paris, Barcelona or Madrid, Berlin or Munich.