It's a very sad state of affairs, and though I can't support what she did, it would take a heart of stone not to be moved by the circumstances. It comes across as heartfelt, and though it is perhaps a little long nevertheless at least to me, in its current state, it had the impact of making me feel very sorry for your aunt's position. Personally, therefore, I would be inclined to send it as it is, since it did the job.
Realistically - and I wish this was otherwise - TfL takes a particularly severe line on cases like this, particularly fraudulent use of Freedom Passes. So unfortunately you probably will need to get over the bad news that TfL is likely to prosecute, and quite likely to want an in-person court appearance. Now clearly you need to try to avoid that, and your letter does this. If there is a court appearance then it is possible that the Court will be moved to keep the sentence at the lowest end of the scale, she won't go to prison, her children are not at threat of removal, it's just a fine. However the Court will sentence according to a criteria and there really isn't any getting around this.
Some London Councils provide help to carers, such as the Taxicard scheme. Though a Freedom Pass cannot be used by carers, it can be the gateway to the council's arrangements.