Debster:
I'm afraid there is not a simple answer to this very difficult problem. The first thing to establish is whether the bill is based on actual meter readings or estimates. Sorry if you said but it was a loooong post. Then since its an unusual tariff you need to be confident that the full and cheap rate readings aren't being mixed up. You haven't provided enough information to advise.
Although the worst of the winter is over its still worth getting a handle on your consumption. Read you meter(s) daily for a week, weekly for a month and then monthly. Get familiar with which reading is which.
Given the high bill and the ineffective heating its likely that insulation levels are poor. You mentioned rental. Is the landlord aware how poor the insulation is?
I don't know the unit prices for Superdeal though I know roughly how it works. Given the lack of competition for this type of tariff it may be that conversion to standard Economy 7 could be worthwhile. Needs an electrician and possibly a meter change, but not impossible. Something for your landlord to consider.
The amount you are paying for energy wouldn't necessary be out of place in a warm gas heated house but I understand your feelings about paying that to freeze.
Start by comparing the unit prices on Superdeal with Economy 7 to see if modifying the system to Economy 7 would be an option for the landlord. Think insulation. There are grants available for the owner.