Princess Luise of Prussia was the second daughter of General-major Prince Ferdinand of Prussia (1730-1813) and of his wife and niece Pss Luise (1730-1820), née Margravine of Brandenburg-Schwedt. Ferdinand was an younger brother of King Friedrich II the Great and uncle of his successor King Friedrich Wilhelm II.
Luise had many siblings: Friederike (1761-73), Heinrich (1769-73), Christian (1771-90), Louis Ferdinand (1772-1806; a general, left natural offspring), Paul (b. & d. 1776) and August (1779-1843, another general, whom left about 11 natural children).
I believe her marriage (1796) to Prince Antoni Henryk Radziwill, from that grand Polish princely family, was made to facilitate the diplomacy during the divisions of Poland. But in fact it turned out to be a love match

…Happy them…They produced 4 sons and 2 daughters I believe.
Antoni was a great art protector, musician and composer, and received people like Goethe, Beethoven and Chopin (who composed some master pieces in his honour). Politically he had no luck, he unsuccessfully tried to linck the Polish to a Prussian alliance instead to Napoleon, later on he was made Prussian governor of Posen (Poznan) and tried to bring together the Prussian authorities and the Polish subject to them, but he also failed.
About her relationship with Queen Louise (why flamboyant?) I know nothing.