Welcome to the forum. London bus routes have always been something of an obsession with me even though it's many years since I lived there, but I keep abreast with developments from afar.
I'm not quite sure why you think Russell Square is specifically in need of a connection to East London, perhaps Limehouse as you suggest. The 205 route that was instigated a few years ago at the time of the introduction of the Congestion Charge provides a useful link from Euston, a short way north of Russell Square, to Mile End and Bow Church in East London, a little north of Limehouse, granted, but not so very different as to require an entirely new route, I would have thought. What the 205 would benefit from imo is a garage allocation from each end of the route to even up the service during times of bad traffic congestion, but this would not be possible under the current tendering system.
Mention of Limehouse, though, leads me to my own thoughts of a gap in the bus network these last few years, namely a connection from London Bridge to points east of Aldgate down the Commercial Road as far as Poplar. This was provided for decades by the 40 through from Camberwell and Elephant, now cut back to Aldgate, and is a journey difficult to do by other means of public transport, especially in peak hours. If re-introduced, perhaps the 15 could be pushed on once more as far as Canning Town from Blackwall Tunnel and bridge the comparative bus desert between those two points these days.
Elephant and London Bridge also used to have a direct connection to Bow and Stratford in the former route 10, but with the Jubilee Line now it might be too much to expect a revival along those lines!