This caught me off guard as well. At least in Midwestern American English, you're just as likely to hear "She eats no meat." as "She doesn't eat meat.", and provided this was the intended meaning from context I'd actually think it makes more sense to translate it as 'She eats no meat.' because that preserved the syntactic structure of the original.
Generally, k is pronounced /ɕ/ before i, e, ö, ä and y. However, we have a lot of loanwords that makes it a bit harder. For example, "att köra" (to drive) is pronounced with /ɕ/ while "en kör" (a choir) is not. Other examples of this are "keps" (cap) and "kö" (queue/line).