Why does it say "Nachon Libi"?
Rashi: My heart is faithful with You 1 .
Radak: Even though I am in Galus for many years, I did not despair from Ge'ulah.
Malbim (57:8): This is when Sha'ul entered the cave, and David's men incited him to kill him. David strengthened himself against his Yetzer, and said "Nachon Libi Elokim" - my heart rules over my Nefesh. It is proper to overpower my Yetzer; I will not strike him.
Rashi (60:8): Both [when You conduct with] Midas ha'Din, and [when You conduct with] Midas ha'Rachamim. (There, the verse repeats "Nachon Libi" - PF.)
Why did he say "Elokim"?
Radak: He calls to Him.
Why did he say "Ashirah va'Azamerah"?
Radak: I trust that You will redeem us, and then I will sing to You. Ashirah - vocally; va'Azamerah - with a Kli.
Malbim (57:8): I will sing that my heart stopped me from killing.
Why does it say "Af Kevodi"?
Rashi #1: My honor is that I sing to You.
Rashi #2: Even according to my honor, I will not honor myself and refrain from singing to You.
Radak #1: Also my Nefesh, the honor of the body [will sing].
Radak #2: "Urah" (the start of the next verse) applies also to Kevodi. Also above (57:9) it says "Urah Kevodi." (There, Radak explained that he arouses it to compose a Shir.)