The German word for "dragon" is der Drache, but the word for "kite" is der Drachen, with an -n at the end. Since the plural for both "kites" and "dragons" is die Drachen, if the definite article is not mentioned, the only way you can tell which word is meant is from the context. This week's new installment of the TV series Großstadtrevier has a good example:
Er wollte die Küche streichen und Maries Drachen reparieren.
He wanted to paint the kitchen and repair Marie's kite.
Caption 8, Großstadtrevier - Von Monstern und MördernPlay Caption
Er fand überall welche, in Schlössern und Palästen,
He found them everywhere, in castles and palaces,
verhext von Hexen und gefangen von Drachen.
bewitched by witches and captured by dragons.
Captions 28-29, Märchen - Sagenhaft - Die Prinzessin auf der ErbsePlay Caption
It is pretty clear from the contexts above that it probably isn't Marie's dragon that is being repaired, nor that the people found in the castles were being captured by kites!
The singular genitive forms are different, however, with "of the dragon" written des Drachen and "of the kite" written des Drachens, with an -s at the end!
Wir haben einen Garten des friedvollen Drachen.
We have a "Garden of the Peaceful Dragon."
Caption 18, Das Tollwood-Festival - BAP und Clueso in der Musik-ArenaPlay Caption
Visit Yabla German and search for examples of der Drache and der Drachen as used in a real world—or perhaps a purely imaginary—context.