I Can't Tell If 〜かどうか

か alone, which you already met for embedding an open question like 何か食べたか, only works when there’s a question word involved. かどうか is for closed yes/no questions instead, “is it or isn’t it,” “will they or won’t they,” with no question word at all.

な-adjectives and nouns drop だ before かどうか, just as they do before plain-form か: 元気だ becomes 元気かどうか, not 元気だかどうか. い-adjectives and verbs attach directly with no change at all.