BJT operation and conservation of charge
My question is Where am I going wrong?


I've redrawn your circuit







simulate this circuit &ndash; Schematic created using CircuitLab

so that is clear that all of the emitter current is through the base-emitter source.  

Remember there is only one base terminal so the way that you've drawn the circuit, with the voltage sources connecting to 'opposite sides' of the base, is misleading you - the base and two voltage sources connect to a single node as I've shown. 

The '90 electrons' from the collector flow through \$V_2\$ and then through \$V_1\$ joined by the '10 electrons' from the base.