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Playing with a broken stick is not a criteria for review. The ref's call on the ice is final. He could have disallowed the goal and assessed a penalty.
It was a very quick series of events and he dropped the stick immediately when it fell apart. The ref had the perfect view of the play. It also looks like there was no immediate protest from any of the Blues on the ice either.
By the strict letter of the law, that probably should have been disallowed, but in the spirit of the game, I have no problem with it being allowed. It happened so quickly that he didn't really "play" with the broken stick.
If that same situation had occurred anywhere else on the ice or at any other time in the game and not resulted in a goal, it is not likely that he would have received a penalty for playing with a broken stick.