Statistically speaking, everyone in the world is a descendant of King David.
There's several claimed paths that seem feasible, even if they cannot be proven.
For one thing, it was common in many European countries in the Middle Ages for Jewish converts to Christianity to be formally adopted into noble families. Many of those conversos went on to marry other nobles. There's some indication from records that various kings of Spain were aware of having some Jewish ancestry, and were particularly sensitive about that topic.
Queen Elizabeth II is one of the few persons in the world that has her own family archive and her own staff to research her familytree full time. She really knows who is family and who is not. In the end the complete royal families in Europe are married with eachother and so. The whole point is if she will as the head of the church of England will accept King David.
It is wise not to open that can of worms.
Looks like the line goes through Paloma ha-Leví Benveniste I am not seeing any strong evidence for this connection. Looks like it was cut in the past but someone added it back in. The connection to Paloma's parents comes from the works of David Hughes and Jaim Harlow, both known genealogical frauds, so ill go ahead and cut the connection again and lock it down.