Mike is the victim.
If Mike's sister and Jack's sister are the same person, then (a) reads "Lily argued exactly once with (Mike or Jack)" and (b) reads "Lily argued twice with (the same Mike or Jack)". These two statements contradict each other. Therefore Mike's sister must be different from Jack's sister.
If this is the case then there are two possibilities:
1) Jack and Carol are brother and sister. Mike and Lily are brother and sister. Jack is married to Lily and Mike is married to Carol. If this is the case, then (a) reads "Carol argued once with Mike" and (b) reads "Lily argued twice with herself", leaving Jack dead. Since Lily did not likely argue with herself, this is not the solution.
2) Jack and Lily are brother and sister. Mike and Carol are brother and sister. Jack is married to Carol and Lily is married to Mike. If this is the case, then (a) reads "Lily argued once with Jack" and (b) reads "Carol argued twice with Lily", leaving Mike dead. Since there are no contradictions, this is the solution.
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