en_tn_condensed/isa/62/01.md

For Zion's sake I will not be silent, and for Jerusalem's sake I will not be quiet

Both of these statements mean the same thing. Here "Zion" and "Jerusalem" both represent the people who live in Jerusalem. Alternate translation: "For the sake of the people of Jerusalem I will not be silent" (See: rc://en/ta/man/jit/figs-parallelism and rc://en/ta/man/jit/figs-metonymy)

I will not be silent

It is most likely that "I" refers to Isaiah.

until her righteousness proceeds brightly, and her salvation as a burning torch

Both clauses reassure the people that God will eventually come and save the people of Israel and that it will be as apparent as light is. (See: rc://en/ta/man/jit/figs-simile and rc://en/ta/man/jit/figs-parallelism)