What is "Sichi"?
Radak: It is the Shir, thanks and praise.
Malbim: It is my song for my life and extra (good that Hashem bestows on me), and not for the destruction of Resha'im. Hashem would not be pleased with a song about the latter - 'My creations are drowning in the sea, and you say Shirah 1 ?!' (Megilah 10b)
Brachos 10a says that David sang Shirah over the downfall of the wicked! Mishnas R. Aharon 3, p.4 - Bnei Yisrael sang Shirah [with Ru'ach ha'Kodesh] while the Egyptians drowned! Shirah of the angels merely praises Hashem; they are "Omdim" (they do not grow). Hashem did not desire such Shirah at this time. Shirah of people for a miracle strengthens their Emunah and perception of Hashgachah, and fixes it in the Nefesh, and is a remembrance for all generations; this is obligatory!
Why will he rejoice in Hashem?
Radak: Hashem rejoices b'Ma'asav, and we rejoice in Him. We will be a nation to Him, and He will be Elokim to us.
Mesilas Yesharim (Perek 19): David reached a great level of knowing Hashem's grandeur, to enter inside, and his heart rejoiced.
Malbim: Even though I will not sing about the eradication of Resha'im from the land (refer to 104:34:1:2), in my heart I will rejoice in Hashem due to this -"Simchah la'Tzadik Asos Mishpat" (Mishlei 21:15).
