Hey smalltown,

A couple things here:

1) The big key in this sentence is the sequence of events given by the verbs. In the original, "might not meet" is future tense but we're looking back and proposing a different history, so that's wrong. So is the fact that you "might not have been here" now...that's using a past rearview when clearly it should be talking about today. So the corrected sentence fixes those errors and gives a logical timeframe: "If mom had not attended, she might not have met dad, and I might not be here now." So the real key - what this question is specifically testing - is the verb order that signals sequence.

2) As to your direct question, "might" still works here because who is to say that the ONLY way this person's parents would have met is through LSE? They might have still met at one of the concerts they attended together or they might have met through a symposium she might have attended while going to a different school. The author is expressing doubt but not certainty as to whether they wouldn't have met.

"Would" still works here, too, but that would mean that that the author is certain that LSE was her parents' only chance of meeting. "If she hadn't attended LSE she would not have met my father and I would not be here today" means that she's fairly certain that the mother wouldn't have ever met her father in any other circumstance. In the sentence as given, perhaps the mother lived in London or would have attended a different London-based school; in the "would" option, perhaps she lived in Tokyo or Buenos Aires and the only reason she ever would have been in London is specifically for that school.

Note that both sentences would be correct - neither is an illogical meaning, but they do mean slightly different things.