My answer is December 13.
Reasons:
Joseph was told the months and Roi was told the dates. (Jan 14, Jan 15, Jan 18, Feb 16, Feb 17, Feb 13, Feb 15).
And Joseph replied: I do not know when Sarit's birthday is but I know Roi does not know either.
This means they didnot have clue til now.
(March 13, March 15).
Roi replied: At first, I did not know Sarit's birthday was but now I know.
Which means Roi know the date now i.e, 13.
(December 13, December 14, December 16)
Joseph replied: So I know when her birthday is as well.
Which means December 13 is Sarit's birthday.