I don't spend this kind of money on holidays, but I have to say I have massively enjoyed my travel in Sweden last year. It was lovely temperature-wise, had the most stunning views, amazing sea swimming, gorgeous national parks for walking in, great public transport and a fantastic variety of international food. I was staying in Malmö, which is in the south of the country so a bit warmer weather wise and incredibly easy to get to from Copenhagen airport. I think you could find something that met your requirements of "stuff to do but not so much as to be exhausting" there quite easily.
I think I would probably struggle to spend £18k, but I guess a nice hotel etc would probably add up.
I tend to go for Airbnb because I prefer to have the space. These are towards the outskirts of the city because they are larger - I would probably go for something closer to the city but smaller myself, but if you have plenty of budget you could just get a taxi.
E.g.
https://www.airbnb.co.uk/rooms/885557521874401557?viralityEntryPoint=1&s=76 is about £3.5k
And then flights on those days look like they'd come to about £1k, but I suspect if you fiddled with the holiday dates you could get that down by maybe 1/3.
The public sauna is wonderful (not sure if it's suitable for children) and there are beaches, places where there are piers for diving, fabulous food, bike rentals across the city. I went to a wonderful national park which was about an hour on the train and then maybe 20 minutes on the bus. The spring water there was the most delicious I have ever tasted.