OK Dwsi I've had a change of heart because it's Christmas and you're so near anyway so it's:-
Let's recap
It's not a winter building, so what is it? = It's a Summer House
It is built on Wales' most northerly point = Llanlleiana Head
It was built by the Stanleys but for the wife's use! = The landowner had it built for his wife
You come across it on Section 10 Amlwch Port to Cemaes of the Anglesey Coastal Walks