Learn Danish: Kvinden benægtede at hun kendte rygtet. - The woman denied that she knew the rumor.

This is because of the sequence of tenses in reported speech. Since the main verb benægtede (denied) is in the past tense, the verb in the at-clause must also shift to the past tense (kendte). This matches English perfectly: we say 'she denied that she knew', not 'she denied that she knows'.

Could I use vidste instead of kendte for 'knew'?

No, because you are talking about knowing a specific noun (rygtet). You use at kende when you are familiar with a person, place, or thing. You use at vide when you know a fact, which is usually followed by a question word or a whole new clause. For example: hun vidste at han løj (she knew that he was lying).

How does the word rygtet get its ending?

The base word is et rygte (a rumor), which is a t-word (neuter). To say 'the rumor', you simply add the definite ending -t to the end of the word because it already ends in an 'e', making it rygtet.

Do I have to include the word at in this sentence?

No, it is optional. Just like in English where you can say 'The woman denied she knew the rumor', in Danish you can often drop the conjunction at. You will frequently hear Kvinden benægtede hun kendte rygtet in casual everyday speech.