Leicester would benefit from 4-tracking, especially from Wigston to Leicester station.
The main beneficiaries of electrification need the Birmingham line to be wired. That seems further down the list. You could have EMUs on the Ivanhoe line, and maybe a local to Derby, but that does not really create the benefits now the 810s exist. This also needs there to be more capacity on Birmingham to Leicester and I am not sure that can be done. Perhaps a local service from Nuneaton to Leicester to give a second train per hour at the stops on that section and extend the current Birmingham to Leicester to Nottingham?
I would be inclined to run London to Sheffield like London to Norwich. That might not be "InterCity", but it would be suitable for the line length and offer the requisite shorter dwells and acceleration to allow more stops south of Leicester. That would need electrification.
So the priority to me would be getting the layout at Leicester sorted and laying tracks in a way that would accommodate electrification at a later stage.