Louis Quinze and Wilhelm II are relevant to the styles of furniture in the room, though perhaps not in a way some readers might expect. Louis XV's name is connected with one of several styles of furniture developed in France because several French rulers (or those around them, e.g. Louis XV's mistress Madame de Pompadour) were patrons of the arts. It is no coincidence that the eighteenth century is sometimes described as the golden age of French furniture. This patronage led to the creation of styles of furniture that are still collected today.
Wilhelm II, by contrast, who was roughly a contemporary of Saki's, is not remembered as a patron of the arts but for his militarism and belligerent foreign policy. That does not mean that no great style of furniture was developed in the German-speaking part of Europe; Jugendstil also had its influence on furniture style, but has nothing to do with the Kaiser. When Reginald mentions Wilhelm II, he is definitely not thinking of Jugendstil. (Speaking more generally, German pre-twentieth-century furniture has never been as attractive as French furniture. Jugendstil, which started in the 1890s, is not mainly remembered for its furniture.)
With this in mind, Reginald's comment about the furniture means that the Princess tried to furnish the salon with stylish (and presumably collectible and expensive) French furniture but also had to compromise and settle from some contemporary, totally unremarkable and cheaper pieces of furniture.