How about this:
I understand the seriousness of the situation and I am committed to resolving the matter.
Upon arriving at Flitwick, I had accidentally bought the wrong ticket to Harlington for the train I was supposed to be on (this is what I used to get through the barrier) - this was a genuine error. I sometimes travel to Harlington meeting friends.
It was after the barriers that I had realised that I had the incorrect ticket, so I immediately bought the 6:59 ticket to London at 6:43, as the 6:43 ticket was unavailable to purchase - I could not purchase as the 6:43 was running late. The reason I boarded the 6:43 train with the 6:59 ticket is because the tickets are flexible and can be used at any point in the day, therefore the ticket was valid for the journey when I boarded.
I must express I did not enter the train with the intention of not paying, I have reference to my purchases if needed.
I would say something like this.
I am writing in response to your letter dated XX XXXX.
Upon arrival at Flitwick station I purchased a ticket to Harlington and passed through the barriers. This is a journey I regularly make for leisure purposes. As I passed through the barriers I realised I had inadvertently purchased the wrong ticket as I was travelling to St Pancras. I immediately purchased a ticket from Flitwick to St Pancras to regularise the situation.
The ticket from Flitwick to St Pancras was purchased before I boarded the train. You will see from the enclosed booking confirmation that the ticket was purchased at 06:43 and I travelled on the 06:43 service which was running late on the day [if you can find out exactly how late it was and state that here].
As I held a valid ticket for my journey, which was purchased before I boarded the train, no offence has been committed and I hope you will confirm that this matter is now closed.
Do make sure that you can absolutely show to them that the ticket was purchased before you boarded the train.