The house of Judah refers to the southern kingdom, with its capital in Jerusalem. Even though Hosea is writing to the northern kingdom, he, like other prophets, does occasionally include their kin across the border. This happens frequently enough to suppose that these additions are original to the prophets and not later editorial remarks, though in some cases that is possible too.1
It is widely acknowledged that the events described in 2 Kings 18:1 – 19:37 and Isaiah 36:1 – 37:38 are when God took mercy on Judah according to this promise. It was in 701 BC when Sennacherib and over 185,000 Assyrian soldiers surrounded Jerusalem. Hezekiah, the king of Judah, had one life and death choice to make. Trust in the Lord God or trust in his bows, swords, and horses to save them. He called out to the Lord for deliverance (Isaiah 37:20). Without having to lift a sword, the battle was won. An angel of the Lord put so many Assyrian soldiers to death in their sleep, that Sennacherib had no choice but to return home defeated.
7 But I will have mercy on the house of Judah, and I will save them by the LORD their God. I will not save them by bow or by sword or by war or by horses or by horsemen.”