"I am not in trouble."
Shouldn't simply こまりません also work? Or am i missing some nuance from their word choice?
I think I'm definitely missing the same nuance as you, but my guess is that this is a case of, uhhh, I'll call it past-tense-continuance, like how "I am hungry" is literally "my stomach became empty". Maybe こまりません has an implication more like "I'm not getting in trouble (or becoming troubled?) right now."
Plugging it into Google Translate (with kanji: 困りません) gives: "I do not need help." Not sure if that supports my hypothesis or not, 'cause, y'know, Google Translate.
That would mean roughly, 'I don't/ won't worry.' (or be bothered or however you wish to translate it) Basically that's present or future tense, but what you want is present progressive. Think of it like, 'I an not currently worrying.'.
So, basically, in English it's Present Simple, while in Japanese it's continous?
こまって (困って) you got, but I think いません is usually kept in hiragana...
Though 居ません brings up "I am not here" in google translate, 居ます seems to be translated to "I'm home" with 居 (きょ) as "living". So perhaps there is some preference in the kanji being used?