וְנָתַתִּ֛י מַפְתֵּ֥חַ בֵּית־דָּוִ֖ד עַל־שִׁכְמ֑וֹ וּפָתַח֙ וְאֵ֣ין סֹגֵ֔ר וְסָגַ֖ר וְאֵ֥ין פֹּתֵֽחַ׃
Rashi on Isaiah
the key of the House of David Jonathan paraphrases: The key of the Temple and the government of the House of David.
Ibn Ezra on Isaiah
And the king of the house of David, etc. From this verse we learn that the house (ver. 15) means the royal palace.