I think the confusion here is not stemming from a difference between German and English grammar. It is instead that the verb itself has a different meaning. In German it is the subject of the sentence that is in a state of being missed by him, where as in English it is the subject of the sentence that is committing the action of missing her. I hope that is correct and makes sense. Please correct me if not.
This made more sense to me when I saw "to lack" as a translation of "fehlen" (http://dict.leo.org/ende/index_de.html#/search=fehlen&searchLoc=0&resultOrder=basic&multiwordShowSingle=on) -> "She is lacking to me". That, at least to me, seems less ambiguous about the direction of "missing" than "to be missing".
I hate this verb (not really, but it's hard to grasp for an English speaker). I just have to think of it as meaning "to be missing from," and 'from'='von,' which is a dative preposition, so it takes 'mir' instead of 'mich.' The same with gefallen and gehoren ('to be liked by' and 'to belong to'- 'bei' and 'zu' are also dative prepositions so they also take mir or dir etc.).
I am getting confused with all with words for missing. I get verpassen is to miss something - like lunch ( i remember it as is pass it (accidentally) and then there was vermissen which means you miss something - as in you long for it. But now theyve introduced fehlt. Googling it it seems it means - to be missing. but here it is used in the same way as i thought vermissen would be?