# He will not break any bruised reed; he will not quench any smoking flax

"He will be kind to people who are like a bruised reed, and he will be gentle to those who are like a smoking flax" 

# bruised reed

"damaged plant"

# he will not quench any smoking flax

"he will not put out any smoking wick" or "he will not stop any smoking lamp wick from burning." A smoking flax is a lamp wick after the flame has gone out and when it is only smoking.

# until he leads justice to victory

"until he makes justice have victory" or  "until he makes everything right"