Reactors can be a great addition to nitrate/phosphate battles.
I would do large water changes along with manually removing the problem during those water changes. That in tandem with the addition of something like GFO
can bring the issue under control. It takes time to get there, but patience will get you where you want to be.
Though you say the nitrates are high, it might be even higher if the amount of hair algae is enough it can be lowering that number. It is common to see false readings on both nitrate and phosphate at times, hence why people get confused as all parameters read 0!
Another part to this battle would be the feeding schedule. If you are feeding corals, I would stop. Liquid coral foods usually don't do much at all and are basically liquid phosphates. This easily breaks the tank out in algae, even I have done it on more than one occasion. This is why I refuse to do it anymore. For feeding your fish, unless you have something with really specific feeding needs, there isn't any reason to be feeding sparingly every other day, or even every three days. I feed my 55 mixed reef tank every 3 days either 1/2 or a full cube of mysis shrimp (rinsed off as the mass produced stuff is packed in what is also basically liquid phosphates) with no ill effects and all happy fish.
For additional reading pleasure, there are articles in our articles section about battling algae. Both of which are in my signature for easy discovery.