Why is David mentioning what happens after death?
Radak: If I die, I will not mention You or thank You in my death. If You will heal me, I will thank You in front of everyone. This is like Chizkiyah said "Ki Lo She'ol Todeka Maves Yehaleleka" (Yeshayah 38:18), and David said "Lo Amus Ki Echyeh va'Asaper Ma'ase Kah" (below, 118:7). After the Neshamah leaves, the body is like an inanimate rock; it descends to She'ol, but the Neshamah rises, thanks and praises without ceasing. A Tzadik desires to live to do Hashem's will while he is alive, to increase the reward of the Neshamah in the world to come.
Why is there no mention of Hashem after death?
Malbim: Mention of Your name is for the existence, for You make exist and create. It is not for absence and void, and not for death - 'Hashem does not designate His name on evil' (Bereishis Rabah 50:8). What is missing [from the world] does not receive Your good and Chesed. This is the purpose of the creation that You created, to benefit those found [in the world], that they should praise You.
