I'd use automotive trim adhesive. It's available at WalMart and any auto parts store. Here's one link:
https://www.walmart.com/ip/3M-Auto-A...SABEgJ0gfD_BwE
You mentioned the LED strip being replaced once already, so you don't want to use anything "permanent" since you may need to remove it for future replacement if more LED's stop working.
If you do use this type adhesive, apply strips of masking tape on both top/bottom of the existing location, pull the LED strip away, clean it as best you can, apply the adhesive (per instructions) and reapply. Once done, remove the masking tape. That way you won't have "over-brushed adhesive" to clean up.....