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".
I don't use House introductions and am lost as to whether and why to reject or accept cheng tuq ghurlaq jIH. It makes logical sense: the speaker is ghurlaq, what sort of ghurlaq? Gurlak of the Cheng House, so cheng tuq ghurlaq. If it's a specific and restrictive idiom we should probably have more examples, and if it's just a little odd, we should probably have more acceptance.
The sentence entered into Duolingo seems to be a bad sentence. Here's a link to a copy of the original MSN posting by Dr. Okrand: http://klingonska.org/canon/1997-09-01a-news.txt
Given that matlh says there are multiple ways and mostly talks about formal introductions, I am fine with cheng tuq ghurlaq jIH, but I am extrememly doubtful of ghulaq cheng tuq jIH that we have entered. I say we delete this sentence from the course and consider adding ghurlaq jIH 'ej matay' jIH cheng tuq je to the Love Skill (where tay' is introduced) with a Best Translation of "I am Gurlak of the house of Chang" and allowing cheng tuq ghurlaq jIH on the reverse translation.