"you who have kept your promise to your servant David my father." To keep a promise is an idiom that means to do what one has promised. Alternate translation: "you who have done what you promised your servant David my father" (See: [[rc://en/ta/man/translate/figs-idiom]])
The phrase "with your mouth" may emphasize that God himself spoke. It was not someone else who made the promise. Alternate translation: "you yourself spoke"
The hand is a metonym for the power of the hand. Alternate translation: "and, by your power, have fulfilled what you said" or "and by your power have done what you said" (See: [[rc://en/ta/man/translate/figs-metonymy]])