I agree with PP, given the situation you are now in - you do nothing further now to contact her. However I would spend a few minutes writing a statement of what happened, with dates/times/key points spoken.
The bind for you, though, is that you do not have a resignation note, so she could easily turn round and say you have merely taken her off the payroll giving no notice, and claim unfair dismissal. To that extent, I disagree with PP. Its your word against hers. Hence the need to write notes on what happened, which you can drag up later if the time comes. Sign them, date them, and get a co-director or senior manager of the business to witness the your date/signature. She probably won't have the wit to do the same, if her version of the 'event' is different. Then at least you've got a clear statement of your side, written soon after the event happened.
The clever way to do this would have been to refuse to issue the P45 until she gave you a letter confirming she wished to leave - which you could have easily done. However to write to her now, after the event and after the contract is demonstrable as having come to an end, in my view alerts her to the potential for a claim against you.
I agree, her motivation is more to do with getting the money from the DWP / freeing her debts.
Unfair dismissal claims have to be lodged within 90 days. Keep counting them down.