"Wine makes me sick."
Translation:Vinum me aegrum facit.
Mihi is the dative of ego, the case you use for the person to whom something is given. In "wine makes me sick", "me" is the person affected by the process, that is to say the direct object, so you need to use the accusative me. In English you use the same form, in Latin the cases are different:
— he makes me sick: me aegrum facit;
— he gives me wine: mihi vinum dat.