There is no new king because the old king died without heirs. In this case, Sir Ector, Wart, and Kay travel to London to participate in the tournament that will choose the new king. In London Wart he finds a sword for Kay stuck in a stone.
We can arrive at this answer because:
The question refers to the book "The Once and Future King."
In this book, we can see that the King of England died without leaving any descendants who could assume the throne.
In this case, it was decided that whoever managed to remove a sword that was stuck in a stone would be the new king and a tournament would be made for that.
Sir Ector and Kay want to participate in the tournament, so they go to London with Wart, a servant.
Upon arriving in London, Kay realizes that he left his sword at home and orders the Wart to get him another sword.
Wart finds the sword stuck in the stone and decides to take it to Kay, he pulls the sword with some ease and this makes him the new king of England, King Arthur.
