You'd need to check several things, firstly if a electricity provider incorrectly bills you after a certain amount of time they can't ask for the extra money afaik, I can't remember how long that is though, I think 6 months is possible but might be a lot longer. So you need to look into that, essentially the elecy company might not be legally entitled to some of the leccy you used as they billed you incorrectly for too long.
Secondly, check your agreement with landlord, is there any stipulation for things like you informing him of problems, or him being liable for checking appliances/general things around the house every X months.
IE if it says you are to inform him of any problems and will be liable for any costs associated with a delay to telling him, you've got no where to go, if it says he's got to make an inspection every 3 months AND he hasn't done so, then the responsibility would fall on him.
I'd give the leccy company a call, tell them you had a broken boiler and ask for all the meter readings they've taken for a couple years, then you might be able to see where the spike in usage came from, if there was any that is.
Secondly, you're on the wrong tariff with completely the wrong thinking.
Not using storage heaters costs you a LOT more than using them, the whole point of economy 7 and storage heaters is to fill them up cheap overnight rather than using electricity at 4 times the cost throughout the day.
You're previous tennants probably did what they were supposed to, boiler barely on, storage heaters filled at night and thats why they had low usage.
What rating is the boiler, it could easily be a 3-4kw boiler, run that for 3-4 hours a day at 26p kw/h thats where your cost likely came in.
You could also ask a plumber/builder to have a look at "guesstimate" if the boiler was faulty for long or not, if theres any warranty on the boiler if its new I really have no experience with that but maybe if the boiler is under warranty then the owner can claim any excess on the bill from them.
The main problem appears to be though, incorrect use of heating, avoiding what you thought were expensive storage heaters, and instead using the much MORE expensive electric heating.
Assuming the boiler is fixed you need to again, get info from the leccy company and see how much you are using now, compared to then.
Theres also the last thing, the estimate might have smeg all to do with the previous tennants, they might have had a £60 estimate yet themselves had a £1000 bill to pay the difference when they moved out. Not checking your own readings is pretty lazy, your landlord and you should have made a note of the reading when you moved in, you should be able to check it now, and you should be able to check it in 3 more days and see roughly if it works out, figuring in that you'll be using a lot less heating than in winter.