I think you now have a letter that you can be happy with - although it might be worth waiting around a bit (maybe until later today) before sending it to see whether anyone else has any comments to make.
As to how to start and end the letter, don't worry too much about this: yours will not be the only letter that is received, and whoever has to deal with it at London Midland will be more interested in the content than whether you have phrased it in the exact way that they like. Personally, if you aren't writing to a named person, then I would suggest that you should start 'Dear Sir or Madam', and sign off 'Yours faithfully', but it's not critical. Given that you have been asked
Just putting your statement and your signature is probably enough
What does matter is making sure that your letter gets to the right person, and that any reply will get back to you. So make sure that
- you send the letter to the right address (if you haven't been sent an envelope, check what is given on the letter they sent you)
- you mention any reference in the letter on the envelope
- your name and address are correct (someone wrote down details when you were first interviewed, and its very easy for things to be written down a little wrong: the Royal Mail managed to deliver this letter to you but if there are mistakes in the name and address you may not be so lucky next time)
- you keep a copy of everything you have been sent, and everything that you have sent (hopefully nothing will get lost. But it's better to have copies of everything and not need them, rather than not have copies and then find you do need them).
Thank you so much for the feedback so far, it really helps! The letter was dated 22nd (Wednesday). If I were to leave it to the weekend to be sent off, would this not run the risk of being delivered late? (Which was why I opted to get it delivered today).
EDIT: Turns out the home printer is malfunctioning, so I guess I can take any further advice, will be sending it off first-class tomorrow.
EDIT 2: Here is the current version
"
Dear Sir or Madam,
On the 9th November, I arrived at my destination station (Snow Hill), and in an attempt to get a cheaper ticket fare, I opted to dishonestly tell rail staff my origin station as being one closer to my destination (Jewellery Quarter). Upon scrutiny, I had admitted to staff I had in fact arrived from a different origin station (Rowley Regis).
My actions were juvenile, cowardly, inconsiderate, and outright disrespectful to the nature of the service London Midland and its staff have provided me over the years for transport. I apologise for the disrespect I showed to both London Midland alongside fellow passengers.
This was a spur of the moment decision and out of character for me, and as such I'm embarrassed by my actions. I assure you that I will not repeat this behaviour.
I can only hope that as recompense for my actions, you would provide me the chance to pay for any administrative costs I have put upon London Midland staff, alongside the cost of the original ticket fare. I would be grateful if you could let me know what sum would be acceptable to you to bring this matter to an end.
I hope that you will accept my sincere apologies and wishes to resolve the matter.
Yours faithfully,
"
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