In the German language itself, the word for "German" is not "German" but "Deutsch." So, "Do you speak German?" is asked, "Sprechen Sie Deutsch?" Therefore, Denglish is most likely a German-born influence upon the English language.
It comes from the "De" in "Deutsch" and the "nglish" in "English" being combined. It doesn't matter whether the native speaker is German or English. "Denglish" is used as a name for a language, so it should be capitalized.
Denglish is typically experienced as German words sprinkled into English text, like spice in stew, to make it exciting. In the USA one might say of an adopted German word, it has been Americanized. In German texts one may find an English word "Eingedeutscht". There is such a market for Denglish, one can even find a comedienne with a "Denglish" stage act.
"Omigod," she says to the überagent."
"Death by Hollywood" by Steven Bochco, page 39 ("über" here means "over", as in "over agent", therefore, "top agent").
"Ich möchte Cornflakes zum Frühstück." My two-year old son declaring in German that he wishes to have cornflakes for breakfast.
In both examples, the person could be said to be speaking Denglish.