I didn't keep an exact count, but judging by the number of stacks of fish and other stuff that I sold, I cast at least 300 times in the Black Market area without catching the Giant Sewer Rat. After that, I took a research break, and read on another site (Wowhead? Warcraftpets?) that fishing from the end of the pier by Cantrips and Crows was the best place to catch the Giant Sewer Rat. So I moved to the Cantrips and Crows, and caught the Giant Sewer Rat on my second lure. Someone else also caught it while I was there. I don't know if the probability there actually is higher or if I just got lucky, but if you're having a lot of frustration in the Black Market, you just might be fishing in the wrong spot.
By the way, I was using the following:
450 Fishing skill
Mastercraft Kalu'ak Fishing Pole with High Test Eternium Fishing Line (+35 skill total)