The letter they got their solicitor to write says the house was 'extremely dirty throughout' which is just ridiculous
Have you Google searched their solicitor ? Do they exist ?
Paying for a solicitor to write a letter remarking that the house wasn't clean, among other things is, frankly, ridiculous.
It's to late to want compensation. The whole thing is odd.
They like a saga, IMO.
I can't see what grounds they have for wanting compensation.
My husband kept cleaning a house we had sold and were moving out of. I was more interested in the moving out process, not faffing around with a dustpan and brush.
In the end I screamed at him, 'the house will not be clean enough for Mrs Buyer. It will only be clean enough for Mrs Buyer when it has been cleaned, probably by Mrs Buyer herself.' (Last name not Buyer, just used this to illustrate my point)
The house was, 'ok' it was, 'good enough'. I guarantee our buyers would have said among themselves, 'the first thing this house needs is a good clean.' That's probably part of the psychology of starting to make the house theirs.
I realise you are upset. Especially as it's an official letter. However, take some deep breaths, let the emotion out of the situation, then when you have calmed down you can reply thus:
Dear Purchaser,
I cleaned the house to a good standard (high standard - your choice of wording). I have enclosed photos as proof - evidence (again, your choice of wording.)
The house was sold as seen.
I do not accept you have grounds for compensation of any sort from me.
Yours truly
now fuck off/jog on - your choice.