A word study analysis of 2 Cor. 7:8-11 in reference to repentance - pertinent Greek words with Strong’s number’s and definitions and number of uses in this passage.
3076 λυπέω - loo-peh'-o - To distress, To make sorrowful (verbs used here 6x)
3077 λύπη - loo'-pay - - Sorrow - noun; root of verb 3076 (noun used here 2x)
3341 μετάνοια - met-an'-oy-ah - A change of mind (noun used here 2x)
3338 μεταμέλλομαι - met-am-el'-lom-ahee - To care afterwards; regret. This word is rarely used for “repent.” (verb is used here 2x)
278 ἀμεταμέλητος - am-et-am-el'-ay-tos - negative form of 3338 (used here 1x)
8 For though I made you sorry (λυπέω - loo-peh'-o) with a letter, I do not repent (μεταμέλλομαι - met-am-el'-lom-ahee), though I did repent (μεταμέλλομαι - met-am-el'-lom-ahee): for I perceive that the same epistle hath made you sorry (λυπέω - loo-peh'-o), though it were but for a season.
9 Now I rejoice, not that ye were made sorry (λυπέω - loo-peh'-o), but that ye sorrowed (λυπέω - loo-peh'-o) to repentance (μετάνοια - met-an'-oy-ah): for ye were made sorry (λυπέω - loo-peh'-o) after a godly manner, that ye might receive damage by us in nothing.
10 For godly sorrow (λύπη - loo'-pay) worketh repentance (μετάνοια - met-an'-oy-ah) to salvation not to be repented of (αμεταμελητος): but the sorrow (λύπη - loo'-pay) of the world worketh death.
11 For behold this selfsame thing, that ye sorrowed (λυπέω - loo-peh'-o) after a godly sort, what carefulness it wrought in you, yea, what clearing of yourselves, yea, what indignation, yea, what fear, yea, what vehement desire, yea, what zeal, yea, what revenge! In all things ye have approved yourselves to be clear in this matter.
We cannot necessarily assume that the “salvation” mentioned in verse 10 is referring to eternal life. These people were already eternally saved; they needed sanctification salvation. There is no mention of the one requirement for eternal salvation; belief in Jesus. They, as believers, had a sin problem that needed to be rectified. Many of the save/salvation references in the NT refer to some sort of temporal salvation, not to eternal life.  For more on this passage please see footnote 20.
Paul rejoiced, not that they had been distressed, but that they had been distressed to the extent that they repented of their sinful pattern of life (the unequal yokes of 6:14; i.e. unholy associations, et al). They experienced godly sorrow which worked repentance in their lives. They were already eternally saved by belief in Jesus but needed to correct the lack in their Christian service. Paul rejoiced in this.
“The Grace New Testament Commentary” sums it up well:
“Paul is writing of the repentance of people already born again. Salvation here refers to deliverance from the deadly consequences of unrepentant sin, not only in this life, but also before the Judgment Seat of Christ (cf. Luke 15:11–24).”
 For example: Matt:8:24, 25; 14:30; Luke 1:71; Acts 27:31; Php. 1:19; 1 Tim. 2:15; 4:16; et al. Almost all OT occurrences refer to some sort of temporal salvation such as salvation from calamity, enemies, etc.