Some clarity:
The King has private income which derives from the annual profits of the Duchy of Lancaster, as well as from other sources.
Duchy of Lancaster: Blanche of Lancaster was a considerable heiress and married John of Gaunt in 1359. Their son, Henry Bolingbroke, later Henry IV, inherited the fortune from his mother. The lands have been passed down to provide for the sovereign since then.
Under the relevant Duchy of Lancaster Acts, the Duchy is required to charge commercial rates for rent. This is overseen by the Treasury.
Similarly, government bodies such as the NHS are required to pay commercial rates as part of their standard procurement policies. This prevents any possible suggestion of corruption over large contracts, which has been an issue in the past.
Royal Lodge is owned by and rented out by The Crown Estate, not the King.
The Crown Estate