The only difference in your gear and mine is the mainhand (plus I went for spellpower instead of haste so I have a few pieces different such as chest, offhand, trinket), my ping is 250 and I wouldn't say I can't play, I've done numerous heroic dummy runs with numerous priests, some that play in world top 10 guilds and there weren't 1k dps differences. I'd really really like to see wws and not recount screenshot.
Also, the other difference is that you have focus magic and I don't. You said you didn't have toft. I can see that you can get that dps by getting chain toft-ed since 4700 * 1.15 = 5.4k, throw in some lucky rng and I can see 5.
But honestly, over the course of 5 minutes, I can't really believe you can maintain 5.7k dps. Even if you had 0 ping, 10 ms human reaction, if god of rng was licking your feet - no, 5.7 is waaay way way out of line so I'll choose not to trust your recount before I see the wws. And then, if wws shows that everything's legit, I'll gladly bow before you and ask for tips