Here is the answer.
1. SK: if I dont know then Wisely also don't know.
In the list, there are two date that the day is unique, which is 7-Jun and 2-Dec. If the D give to Wisely is 7 or 2, then the M give to SK must be 6 or 12. Since Wisely cannot tell the birthday, the D <> 7 and D <> 2 and M <> 6 and M <> 12.
From the fist conversation, we eliminate some date and remain these:
4-March, 5-March, 8-March, 1-Sept, 5-Sept
Wisely: Initially I don't know, but now I know the answer.
Base on this statement, then D <> 5, because if D = 5, Wisely dare not make the above statement.
Now we have these list:
4-March, 8-March, 1-Sept
SK: Then I also know the answer.
From SK statement, we know that M <> 3 because if M = 3 it can be 4-Mar or 8-Mar, thus we know that M = 9
The answer is 1-Sept.
Do you get it XD