So...
A friend of mine was recently in a similar situation. Said friend came to this forum a fair bit to seek advice [as an observer, not a poster], and said to themselves that they'd post their thoughts online once all was said and done. All is now said and done. This friend may be somewhat vague on some of the details in the below but they're quite clear that the situation of the OP was [sort of] similar to theirs. This friend also knows how stupid they were, and how lucky they were to resolve their issue without a prosecution. This is just opinion of course, and my friend would like to remind anyone seeking advice that there is other, similar advice available on these forums from more experienced posters and that this advice is - of course - incomplete without knowing what's in the RPI's book. Finally my friend would like to say thank you to other contributors to this forum as they read it a fair bit a few months back. Anyway...
Here's the grim bit - because of the false details you are almost certainly going to be put forward for prosecution [by which I mean a court date], barring an extremely well worded letter or intervention from a solicitor. Many times people say 'wait for the letter/summons and act after that' but because you have in this case committed 2 offences it's, in my friends opinion, likely you'll be put forward for prosecution and your day in court, unless you met a particularly kind RPI who chose not to write up your indiscretion. Others may disagree [and I would note it varies from TOC to TOC], but I am speaking using my friend's experience of a similar case that was heading to the court, with a similar TOC.
While you are asked to write on the back of the initial letter from NR, this is not the only correspondence you can send them. They will accept other written statements, or evidence, after the 14 day period in which they want their letter back. If, for example, you feel you need to send supporting evidence then you can do so.
My advice, in the simplest terms, is consult a solicitor. A local law firm will be able to help with the wording of your letter. There are a number of firms that specialize in fare evasion matters who will cost more but have a better knowledge of the prosecutors at the TOCs, and of how to mount a defence on your behlaf. I don't want to sound like I'm doom-mongering, but reading your posts you don't sound like a great wordsmith and i think the language you will be required to use to adequately state your case may need some help from outside. That isn't meant as any kind of insult, it's just advice.
If you choose not to get help writing your letter my advice would be as follows:
If you regularly pay for rail travel then say so, and prove it [tickets, bank statements, season tickets]
State that you are willing to admit to your offence if - as it appears - you are guilty of them, and don't wish to fight it. From your posts thus far I think you stand little chance of fighting it successfully.
If this was GENUINELY a one off incident [ergo you have never been in trouble on the railways before] then say so, and make it very clear that this one off incident won't be happening again. Assuming you have never been arrested or in any trouble with the police - state this also.
If, for some reason, some outside event has caused you undue stress/issues on the day of the incident then state so - HOWEVER - this piece of advice comes with the warning that if you state this to be the case you MUST be able to prove it as well. I.E. - if you were stressed because of 'exams' make sure you can prove you have exams at this time of year, if you were stressed because you were being evicted then make sure you have an eviction notice from your landlord that proves this, if you were stressed because your house burnt down then provide details, a crime reference number and so on. In short - don't just say this if you were chancing your arm and you can't really prove it. You mention you were ill - were you really? Were you SO ill that you weren't thinking straight. Could you prove this in a court of law? Ergo - it's not just a "stomach bug" but a long term health condition/mental health condition that you have documentary evidence for?
I think you're a student. So presumably you are aiming to do something in life that would be made difficult with a criminal record? Again, state this, but you MUST be able to prove it [I.E. A degree or course acceptance certificate]. Also if, for example, you GENUINELY would lose your job if you got a criminal record [again, you need to prove this and not just say it - so a provide a contract that details this fact clearly] again state this. If you lose your job then the public purse may need to step in to provide for you so - it could be argued - your prosecution is not in the public interest.
The message I'd like to send with the above, not just to you but to anyone who - like my friend - might happen upon this post is that just 'saying' these things won't wash. Anyone can say they were stressed, ill, knackered or whatever - you must be able to prove it, and if you can't prove it then don't say it. Or don't try to prove it in a half arsed way as no one will be interested in that. Don't play meaningless sympathy cards that don't stand up to scrutiny.
If you have documentation that proves the sort of things I have mentioned in the above, you may wish to consider submitting them with a letter, or as a separate package. I really want to be very clear on this so will say it again - if you can't prove it, don't submit it. A good example might be credit card statements - if they PROVE regular travel patterns, and regularly paying for travel then by all means submit them. If they suggest broken travel patterns then I'd think twice about sending them in. I'm going out on a limb here, but there's something about the way you bought tickets on this occasion that make me think you may not wish to submit these.
Finally, and obviously, state that you are looking to resolve this matter via a settlement, and that you are willing to pay any reasonably incurred costs for the TOC.
And needless to say, apologise for your never to be repeated mistake.
As a side issue, we all really want to avoid a criminal record of course, but getting one really isn't the end of the world. For most employers it'd be a bit akward and a pain in the arse to state that you have one [which you would need to if asked], but by no means does it close every door previously open to you. That isn't me advocating getting a criminal record btw! More saying that the world keeps turning if you have one.
Good luck.
And thanks again to these forums from my friend. It's a tough place a lot of the time, and sometimes offers advice and comments you may not want to hear, but it's better to be braced for the worst and prepared than it is to think it'll all just be OK.