1 - Did you take any readings on any of the meters yourself prior to any exchange? If you did then you would know if the readings shown fall in line with what you had taken. If the readings shown on the bill are labelled something along the lines of "read by us" then the readings used have been taken by meter readers or metering engineer's. While it is possible that the readings may have been inaccurate say for a mis read or input incorrectly, if all the readings again fall in line this would pretty much rule this out. (your own reading owuld have been useful in deteriniming their accuracy).
2 - Your correct energy suppliers cannot back bill for more than 12 months in some curcumstances. I would contact e-on again, mentioning this fact. It is likely that the first agent you speak to won't be able to answer this but make sure s/he either refers to their billing/unbilled/billing code teams, or transfers you through to them directly so you can have adiscussion with them. They will be able to ascertain what deductions if any are to be made on the balance you have owing.
3 - I very much doubt any supplier in their right mind will be asking for the full balance immediately. They will understand that the sum owed is quite considerable. Discuss payment arrangements with them at the very least you should be entitled to aks for a repayment term to match the length of time this problem has occured for. So for example if the readings have been swapped for 4 years you should be entield to 4 years to pay the balance. (bear in mind your payments will be covering ongoing consumption first and then a payment towards the balance - so changes in price and consumption may require an amendment to the figures you are paying).
You could try asking for a reduction in the balance for full payment. i.e 10% of if you pay the remaining in full, or 5% for half the balance up front and the remainder over a payment term. Suppliers usually consider these options.
Although it is a bit too late now, you have to take into consideration that your supplier will have been relying on the information the meter readers have been providing them. If you at no point had called to inform them the readings were the wrong way round (i.e. your night consumption being billed at the dy ates and vice versa) they are just as much a victim as you are. Obviously of course the impact on your is far far greater than the supplier. This is a prime example of why it is very important every consumer of any product or service checks the infomation you are being sent for accuracy.
If the repayment values are still too high despite lengthy payment terms being offered it might be worth considering swtching your meters for pre-payment meters where the debt balance can be allocated to the meter and repaymetns set as low as £5.00 per week, although any figure higher than this will clear the balance quicker for you. I personally would only take this option as a last resort - and in addition i would ask them to allow you to have a credit meter installed again once the balance is cleared (assuming that to date you have never had a problem with payments). If they agree this ask them to confirm this in writing. I mentioned this as it is becoming common for suppliers to refuse to exchange pre-payment meters for credit meters where they have been installed to recoup a debt, this is to protect their revenue of course.
Additioanlly if you remain on a credit meter, consider setting up a direct debit for the extra discounts, and look at the tariff your on considerswtiching to either a capped plan for stability in prices for however many months the plan lasts or their cheapest available tariff (which is uslaly an online tariff) again to reduce as far as possible the payments you will be required to make.
On a more personal note, while i understand you will quite rightly be very angry about this situation remember that it is definatley not the fault of the agnet/s you will be speaking too. Try keepng calm and not shouting at them. From a personal point of view i tend to be far more inclined to go that extra mile to help customers who don't shout/scream and cast insults than just doing the bare minimum for the ones who do.
The answers I give you may not want to hear. The answers I give may not be easy. The answers I give will be the truth. If your prepared for this - ask away!