Welcome to the forum.
Don't try to write to Chiltern just yet: almost certainly they haven't given you anything with a case reference on, and writing without one will give some poor soul in the offices an almost impossible job of trying to match your email up with your case. As well as that, my understanding is that Chiltern contract this sort of work out to an organisation called 'Transport Investigations Limited' (TIL) so not only will your email have to get itself associated with the right case, but it will have to go to a different office.
When TIL do write to you (which will probably be in a few days or weeks time, but could take up to six months) then that's the time to respond, and explain that you have made a mistake, that you are sorry, and that you will take steps to avoid doing it again. Try not to be deterred by the tone of the letter from TIL - at first glance it will probably look like they have already decided to take you to court, but read it closely and you will see that they are giving you the opportunity to give your side of the story.
@Hadders often comments on threads like this with some excellent advice - you can see an example of it at
https://www.railforums.co.uk/threads/til-child-short-ticket-next-steps.224435/post-5386948. With some luck, TIL will understand that you made a mistake and will agree an out of court settlement. With rather less luck, you may have to ask them a number of times if they will agree to settle out of court, but the chances of resolving this before it goes to court are good.