I am Ghurlak of the House of Chang is what would come most readily to my mind. I am Ghurlak of House Chang also works for me (it sounds very Dune), amd it's accepted. The single "the" threw me though, so I answered I am Ghurlak from the Chang House, which should, I suppose, be accepted. It sounds very suburban American to me--e.g. we do not do that in the Wilson house.
"I am Gurlak of the House of Chang" and "I am Gurlak of House Chang" are both already accepted as responses.
I'm not really au fait with Star Trek, truth be told, so I'm not sure what the most common way is to speak about Klingon houses in English -- I'm willing to change the "default" / "best" translation to what users think would be most common. (For word bank exercises, this should affect which tiles are presented.)
"The House Chang" sounds weird to me in English, I would never enter it as an option. However, I'm not confident enough in all varieties of English to say that it is never said that way in any dialect, so I would leave it just in case. It's another one of those entries that I wish I could mark as "accept, but never show".
The answer that's shown is never really an issue to me; it's only when I am presented with a word bank that has a set of words I'm not expecting.