In the northern part of Hampstead Heath, you find the Kenwood House. Preferably you can take the tube to Highgate station and through the Highgate village to the Kenwood house. The house hosts the 1st Earl of Iveagh´s collection of paintings. There is an impressive collection of works from British painters and other painters such as Rembrandt and Vermeer.
It´s hard to believe this is London. You find lovely paths to walk, hidden places and rolling hills. An act from 1929 stipulates that the house and surroundings should be open to the public free of charge.
Check out The Highgate Hilly Walk.