2:15a “You know that the kingship was mine,” he said.

Then Adonijah said, “You (sing) know that I had the right to be the king.
-or-
He said, “You are well aware that I was ⌊supposed/expected⌋ to be the ⌊next⌋ king.
-or-
Adonijah told her that it was he who had the best claim to the kingdom/kingship.

2:15b “All Israel expected that I should reign,

Everyone in ⌊the land of⌋ Israel was looking/turning to me to be their ⌊next⌋ king.
-or-
All ⌊the people of⌋ Israel hoped that I would become king.
-or-
All ⌊the people of⌋ Israel were expecting him to be the ⌊next/new⌋ king.

2:15c but the kingship has turned to my brother,

But then the kingdom went/passed to my brother instead,
-or-
But that did not happen. My ⌊younger⌋ brother is now the king,
-or-
But everything had turned out differently and his brother ⌊Solomon⌋ had become king.

2:15d for it has come to him from the LORD.

for Yahweh gave it to him.
-or-
for this was what Yahweh chose to do.
-or-
Yahweh had wanted to make Solomon king.

2:15c-d (reordered)

But Yahweh decided that instead my brother should become king.

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