After debating if I should go into the boutique as Wendy B pointed out, and after talking to gal pals, SG found a way to fix the broken LV strap. I simply applied white glue using a toothpick to spread the glue out thinly and evenly onto both insides of the strap and secured with cloth pins. (click here to see part 1)
After about 2 hours, the strap looks fine to me. At least I can use the pochette again running errands or as a make-up bag.
As for hardware discoloration, I applied brass cleaner which removed quite a bit of dirt and the surface became cleaner and shinier compare to before. Even if it's not like new, it's still better than the prior condition which I am willing to live with.
***I don't know if brass polisher can safely clean LV hardware but the result is okay for me. So if you are going to try, test a small area first.