Really, you'd use 何 as part of the affirmative sentence? I would've thought:
今日は何の映画を見るか決めたの
Would've been better in the affirmative: "I've decided which movie to see."
Where is the "which" in your version of the sentence?
Whether the full sentence is a question ("Have you decided...?") or a statement ("I've decided...") the
embedded question is still "which movie to see?". "Which movie to see?" being a Wh-question, it requires the question word 何の~
But why does it matter what the embedded question is if the whole sentence is to be answered by a yes/no answer? Can you expand on your explanation?
I'm not quite sure what you're asking here. It's not a matter of what "matters", I was simply clarifying why your initial interpretation of か marking a yes/no question was incorrect by pointing out that the か is not marking the yes/no question "Have you decided...?", but rather goes with the embedded Wh-question "Which movie to see?"
In fact, the embedded question remains exactly the same regardless of what form the full sentence takes.
今日は何の映画を見るか、Aさんに聞きました。
I asked A what movie she's going to see today.
今日は何の映画を見るか、Facebookに投稿しましょう。
Let's post on Facebook about what movie we're going to see today.
今日は何の映画を見るか、教えてね。
Tell (lit. "teach") me what movie you're going to see.
This should make it clear that the か has nothing to do with the yes/no question "Have you decided...?" in the original sentence, yes?