Welcome to the forum.
I'm afraid the general advice is likely to be to pay the £111.50 and accept it as an expensive lesson. But please let me work through why that's what people here are likely to say.
It's most likely that the £111.50 is made up of £21.50 for an anytime day single from Blackpool North to Manchester Piccadilly, plus £90 for the costs that Northern have incurred. The train fare (£21.50) is calculated on the basis that your railcard-discounted ticket wasn't valid at all (you would have needed a valid railcard, and you didn't have one of those), so you needed to pay for a valid ticket without any account being taken of the money you had already paid for your railcard-discounted ticket.
So where does the £90 in costs come from? It seems like a lot - until you think about what costs Northern has incurred. Someone had to pull you over and spend a little time talking to you. Someone (probably someone different) has had to check the report and prepare the demand that has been sent to you. Both these people will have needed an office to work out of. But let's assume that one person looked at your case, and that they're paid £25,000 a year. If they work 2,000 hours a year for that money - 50 weeks at 40 hours a week - that means that the railway needs to pay them £12.50 an hour. That would mean that someone had spent around 7 hours looking at your case if the costs came to £90.
It's most likely that less than 7 hours was spent looking at your case - but once you add in the costs of office accommodation, as well as things like the employer's National Insurance contribution, you can probably see where the £90 came from.
In short, the £111.50 is a lot, but if the railway was put to it, they would be able to justify it.
It's also worth thinking about what will happen if you don't agree to pay the £111.50. It's most likely that the railway have asked you to pay this rather than going to court. If the matter does go to court, then on the facts as you have told us, I would expect you to lose and the railway to win. That's because the law expects you to make sure that your railcard is valid - getting a new one later the same day isn't good enough.
So if you lose at court, what will happen? You won't go to prison - but you will have to pay a fine. This would be as well as paying the train fare and the railway's costs (likely to be much the same as the fare and costs that they have asked you for to settle things) and as well as paying court costs. In all, it's likely that losing in court would cost you more than the £111.50 that you have been asked to pay now.
On balance (and other people here may have different views) I would advise that you pay the amount now asked for, and close the matter down.
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