The repetition the word mountain is intentional. When Isaiah says that the mountain of the Lord’s house shall be established in the top of the mountains, he is speaking of the ultimate place of authority. It is literally the Lord’s house, which we understand as a temple.
In the ancient world, mountains might substitute as a sacred location to commune with God. Moses went up a mountain. The height of the mountain symbolically touches heaven. The particular temple that Isaiah describes is not only in the top of the mountains, but “exalted above the hills.” That phrase might suggest the temples in other nations dedicated to other gods. They might have their temples, but the Lord’s house is above all. It is a mountain where the others are hills.
The physical location of the temple need not actually be in the mountaintop. It is there symbolically. As the place where God might dwell, it is also the place from which God’s word, and God’s law, will flow to all peoples.
For Isaiah, there was a temple in Jerusalem, and therefore Jerusalem could be a substitute source of authority. The king might be in Jerusalem, but the it was ultimately the temple that provided the king’s authority.