It's not so bad with just mg/kg, but J/g K might be confused. Or should that be J/g/K?
J g^-1 K^-1 is utterly unambiguous (when properly typeset) even when on one line. Once you have the "style" guide dictating that J/g K has superscripts, then mg/kg falls under the rule as well--otherwise you'd have to ask why some have the superscript and others don't.
m/s/s is also "legal" but it's just not a form of notation used. ms^-2 is how it's done. G (the gravitational constant) is measured in units of m^3 kg^-1 s^-2. Try expressing that in slash notation.
x per y tends to be a special case in spoken language and informal writing, but standard notation is still much preferred in publication.
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