From my experience you will have to do the water changes and pray. I have done everything mentioned above, but the water changes and siphoning out the chunks of slime is what does the best. I would recommend falling ALL the advice above, i.e. check water, lights, flow, etc. Next, siphon off as much of the stuff as you can. Then, do a water change of 25% or so, THEN medicate the tank with that chemical powder slime killer stuff. Oh, in case you didn't know, make sure you remove the carbon when you medicate with the slime killer powder. After you treat the tank for a few days, do another water change per instructions on the bag. That is what took care of the slime in my nano tanks. I got lucky and only had to deal with a 12 gallon and a 2 1/2 gallon.
The slime should not harm the corals as long as you keep an eye on it. I guess if you left it alone it could 'smother' some stuff, but all you have to do is direct a jet of water with a powerhead or your hand to knock any off the corals that is on them. I only saw that the slime seemed to cover a few kinds of polyps, like star and glove, most of my other corals didn't seem to be too bothered by it.
Good luck, and a few hermits will eat the stuff too, but it seems to grow too fast for them to handle it all, and they seem to prefer other things more.