What is the meaning of "Hashem Lehoshi'eni"?
Radak: Hashem said that He will save me from the illness, and so He saved me. This is like the Lamed in "Lehargo v'Armah" (Shemos 21:14) and "Lishkav Es Bas Yakov" (Bereishis 34:7. Even though these are infinitives,) he did so.
Malbim: He will save me also in the future.
What is the meaning of "u'Neginosai Nenagen"?
Radak: For the salvation that you saved me, I will always thank You. I and my colleagues will play the musical songs that I composed for the salvation. All our lives we will play them.
Malbim: Refer to 36:20:3:2.
Why did he say "Al Beis Hashem"?
Radak: This is like b'Veis Hashem. Sometimes Al is in place of the prefix Beis, e.g. "v'Al Charbecha Sichyeh" (Bereishis 27:40) is like b'Charbecha, "Al Sal Echad" (Shemos 29:3) is like b'Sal.
Malbim: The songs that I used to play to delight in them, from now and onwards, my entire life, they will be for Beis Hashem. I dedicated my songs and Simchos to Shamayim.