Is "Mishenichnas Adar Marbin B'Simcha" just for the month of Adar, and then you take a step down in simcha for Nisan? Or are you supposed to raise your level of simcha and stay there until Av, when you're supposed to decrease your joy? If it's just for Adar, why? What about the other months in which we were saved in some way (Nisan, Kislev)?
Rashi to Taanis 29a writes that the reason for increasing joy upon the entrance of Adar is that it is a time of miracles, such as Purim and Pesach.
The implication of Rashi is that this simcha should last until Pesach, as that is still the time period of the miracles which cause this to be a season of simcha.
(And perhaps you should sing the jingle until then as well, but I can't guarentee you'll have any friends by the time Pesach comes.)
According to the Michtav M'Eliyahu (Vol. 2 pg 125) on the essay titled "משנכנס אדר" he discusses how the command
to be joyous when Adar enters is related to the joy of Purim.
He explains that the happiness we feel on Purim is a הכרת הטוב - recognizing the goodness - of the miracle of Purim, which is all about revenge. HaShem took revenge on Haman and ונהפוך הוא - turned the tables - and made Mordechai the Prime Minister in his stead.
The joy required to celebrate this הכרת הטוב efficiently requires work and preparation, continues the Michtav M'Eliyahu, and one needs to start this preparation as soon as Adar starts, in order to be in the right mood when Purim arrives.
ומתחילין כבר מראש חודש לגודל ההכנה הנדרשת, וממשיכים בעבודה זו מיום ליום
The Ramchal (Derech HaShem ח"ד פ"ח) says that the point of the joy of Purim is the קבלת התורה - celebrating the re-acceptance of the Torah.
So - using the reasoning of the Michtav M'Eliyahu - we would say that we need the 2 weeks to prepare for it; similar to the 49 days to prepare for Shavuoth.
So - back to the question - once Purim is over, one can wind down and get ready for entering the Pessach mood. Each Chag has its own mood and preparation period.