Both sentences are correct. However, there is a slight difference between the two verbs. If one says he 'met' a girl, it implies that a mere introduction was made. If one says he 'knows' a girl, it implies that he as some more background information or, perhaps, a few passing meetings with the young lady.
'I met her through...' most usually means a singe introduction.
'I know her through...' most usually means several meetings that have been facilitated or in the presence of the intermediary.
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