One thing to keep in mind when reading how I “translate” things: Unless I am breaking down a sentence piece by piece, and then Frankenstein-ing it together to keep the Japanese grammar 100% intact so as to explain a specific point, when I give a translation, it’s already been filtered through English grammar, so it’s more of an interpretation than a direct translation. The reason why my interpretation for that specific sentence sounded more natural is because I have skipped the step of “Here is the exact Japanese structure” translation and gone straight to shuffling things around for the English structure and giving a natural-sounding English sentence.
To give an example, if I were I to “show my work,” so to speak, on the sentence you were asking about, keeping the Japanese structure intact:
見たとこ = short for みたところ, “judging from appearances”
お前に = “you,” with a に particle attached, indicating the verb is being done to this
懐いてる = 懐く, in て form
みたいだし = みたいです, casual form, with a し sentence ender, which is sometimes used to give a reason, the English “so” works fairly well as a similar device in this context
I would consider that one “unit”, and stop there to get:
“Judging from appearances, to you (the cat) seems to be becoming attached, so”
Then I would move on to the next “unit”
悪さ = “mischief”
しない = “to do”, in negative form
よう = I took the “way; method of” meaning, it’s a suffix that turns the preceding verb into a noun
しつけてやれ = しつける in conjunctive て form + やれ in command form, so “do training,” roughly
Second unit becomes, with more twists and turns required than the first:
“(In) the way of not doing mischief, train (the cat).”
Putting both units together:
“Judging from appearances, to you (the cat) seems to be becoming attached, so in the way of not doing mischief, train (the cat).”
Which is really awkward and wordy, but (mostly) retains the Japanese grammar. After that, I just sorta go to what sounds intuitive, and it’s not a far leap to the more natural:
“By the looks of it, it’s grown attached to you. Train it not to misbehave.”
I dropped the extra “seems to be” because in English that “by the looks of it,” or “judging from appearances” covers that, and it’s redundant. I also sort of changed the tense of the “becoming attached to,” to a more past-tense “has grown attached to”, just because that sounds more natural in English. As for why I dropped the “so” conjunctive, honestly, that was just me forgetting about the し, to an extent. We don’t necessarily need the “so” in English, though, since that second sentence following the first sort of implies it, in my head (though that could just be because I’m vaguely aware of the し existing, and thus ascribe that to it. Perhaps out of context, it wouldn’t sound as if it has any implication at all.)
Regardless, that’s the basic process that I would go through when interpreting into English. It was how I broke things down when I first started out, and even now, if I run into a particularly difficult sentence, I’ll do that, but for the most part, nowadays when I read, I’m actually just reading, not deciphering, to use @ChristopherFritz’s terms, and like when reading English, I just sort of “know” what the sentence is trying to say without having to go through that process, though admittedly much more slowly than I comprehend things in English, and I run into far more Japanese sentences that I have to think about than English ones, as well as often having to re-read to catch mistakes (like thinking that Kotoha was the one speaking, just as a recent example).
The best comparison for how my reading feels nowadays I can think of is it’s like reading a highly technical/specialized textbook or research paper in my native English. I know what most of the words mean, excluding some jargon, and I understand the grammar mostly, but when put together while describing a difficult subject, I have to slow down and pay more attention, and have to either look up or guess from context what the jargon or certain words mean. Simple Japanese now feels like highly-complicated English to me, if that makes sense? I’m bad at analogies and explaining myself. Haha.
The tl;dr of that breakdown, by the way, is that the reason why my translation didn’t sound as clunky is because it wasn’t a direct translation so much as an interpretation.