Speaking as a former council tax biller we had a "money saving policy" to not send a bill a student on their own in a house with the zero bill (this was possible on a old computer system but not when we changed systems,as we could supress a bill like if we knew someone had moved it was pointless to send a bill for the new financial year relating to a house they had left!)
Sometimes the reminder to remove the exemption was missed and we then billed someone for the 3 previous years we had missed. It was a whopping bill as she lived in a very large 5 bed house.
It was deemed by management that we wouldn't persue it BECAUSE we couldn't prove that we had sent the zero bill with the booklet stating that if you were no longer entitled to the exemption then you should notify us.
Firstly get the single person discount of 25%.
Deal with this year's bill first as they are under pressure to collect in the current year by 31st March.
You can offer a lower amount for paying back the previous year (although I would be inclined to argue that one and ask to see a copy of the original exempt bill - this would have told you the charge for the year, zeroed out, and stating that you are exempt from paying this charge because you are "a student" ) The computer will show when they sent the bills out (or not in their case) this would be produced early March.
Council Tax is governed by legislation and by law they should have sent you bills despite you being exempt to show you what you would pay if you weren't exempt. The worst they can do is take you to court -its a magistrates court so nothing goes on your credit file like a county court ruling would and this gives them a "liability order" showing that you are the liable person for ctax.
They can at this time request your employment details and take a set percentage directly from your salary. This percentage is fixed and is set by central government. They can take 2 lots at the same time but we only did that to people who hadn't paid us for years as they job hopped etc and paid nothing, zilch, zip for ages.
This is not your fault, it is their incompetance. Ask to see the original bills they supposedly sent and I would ask to speak/correspond with the manager as it will be he/she who ultimately makes the decisions.
Phew that was long.