My two cents..

Cheaper (in terms of both money, effort, and time) to book a completely new flight from LCY?

You'd save :
-a high cab fare,
-having to get right across London
..and through Heathrow,
-hours and hours of time (when LCY has a 10min before departure check in and EI wants 2 hours?)
-sleep

and no, I don't work for Cityjet