Who says "Zeh ha'Yom Asah Hashem"?
Rashi (27, citing Pesachim 119a): Shmuel said this when he anointed David.
Radak: In the future, all will say this in response to the kings' astonishment. It could be only from Hashem, so we should not be sad about Yisrael's salvation. Rather, we should rejoice with them!
What is the meaning of "v'Nismechah Bo"?
Shir ha'Shirim Rabah (1:3): It was not known whether "Bo" refers to the day, or to Hashem. Shlomo explained - "v'Nismechah Bach" - in You, in Your salvation, in Your Torah, and in fear of You; in the 22 (the Gematriya of Bach) letters in Your Torah (Shir ha'Shirim 1:4).
Radak: We (the nations) will rejoice with Yisrael on this day.
Malbim: We will rejoice in Hashem Himself, and not in the salvation. The primary Simchah is that we know that Hashem is with us, and He gave to those who fear Him a banner to raise.