First of all, your dot uptime can be better. Also, you aren't casting dark archangel often enough.
Now looking at the burn phase, I can see that: You are moving too much and not casting enough. You didn't cast enough mind blast and mind flays. Plus, your dot uptime wasn't 100%, which is unacceptable at this critical moment.
You really need to minimize movement, and when you move, spam devouring plague or SWD (during execute).
For comparison sake, I do ~40.8k dps during the burn (and I forgot my flask...) without the legendary, while you did ~35k. I choose to look at this phase because it's the most predictable phase.
You can't rely on a legendary to do decent dps, even bad players with a legendary do bad dps (not AS bad, but still). Looking at your logs, your DoT uptimes seem pretty low compared to my first kill log:
Yours (Fight length 12:15)
Name Amount Uptime
Shadow Word: Pain 19 654.5 s 88.9 %
Devouring Plague 33 579.0 s 78.7 %
Vampiric Touch 46 543.0 s 73.8 %
You're also casting Shadow Word: Pain a lot, not sure if thats a recast due to it dropping off, or because your multidotting the adds. Also, if mana permits, you should be spamming Devouring Plague every time you need to move. I'm not a wiz with WoL so I can't say for sure, but it doesn't look like you're doing much at all when moving, except maybe a Shadow Word: Death. One big thing I can see as well is your Shadowform uptime: 79%, meaning that you were out of it for about 3 minutes. That's a long time, even acounting for hymns and the occasional helping heal.
Looking at your armory; you need to get Darkmoon Card: Volcano. Its the 3rd best in slot trinket at the moment, behind Heroic Necromantic Focus and Heroic VPLC. Replace your rep trinket with it. Apart from that, theres not much more i can advise, other than the basic stuff: double pot, be casting ALL the time, queue your spells so you aren't wasting time between casts and so on.