Jemandem ist schlecht = Es ist jemandem schlecht
In German, if someone is feeling sick, nauseous, not well, you may say "jemandem ist übel/schlecht". This is an idiomatic and shorter way of saying: "Es ist mir übel". "mir" is the dative object/pronoun. The sentence "Meinem Kind ist schlecht" expresses this, but it is not saying that the child is bad.
CraigTP, the difference is simple. When you say something is something else, you make both words nominative, e.g. "My child is bad." => "Mein Kind ist schlecht.". Compare with "My child feels bad." => "Meinem Kind ist schlecht." Like OP said, we use the dative case therein.
And as an extra piece of advice, when you talk about your own body parts and doing something to them, you don't use the accusative to simply point to them, but first tell "for whom you are doing something" with dative: "I wash my hands." => "Ich wasche mir die Hände." or "I broke my legs." => Ich brach mir die Beine."
Could it also be "Es ist meinem Kind schlecht"? The way this sentence (and other equivalent sentences) works in my head is something akin to "It's bad for my kids" (as in "Things are bad for them"). That's also how I internalize "Mir geht's gut" = "For me, it's good" = "For me, things are good"