Purgatory is unnecessary in Mythic+. She is far better off running Bonestorm because, especially on bigger pulls, it's a fantastic CD for both healing and damage. Also, depending on when/how purgatory procs, it's not even guaranteed that you'll be able to rescue her from the proc.
Are blood DKs in a good place for M+? I mean, we're far from being the optimal choice, but they are doable. I'll cite the usual here...bone shield stacks being removed on the ICD due to inconsequential dot ticks is annoying, and the lack of a decent DR cooldown also sucks (need baseline Rune Tap or IBF back), and our mastery is incredibly weak. I ran DHT +10 the other night, and we were only 2 minutes off from finishing in time. We had 2 silly wipes early on in the dungeon that if we clean those up, we'll make it in time. There's A LOT of kiting involved, and it's basically mandatory when the mobs reach 30% and/or when your necrotic stacks get high. Also, AMS must be used to help Necrotic stacks drop, or to prevent super high stacks on larger AOE pulls. The small blood globs in DHT where there are like 10 or so in a pack will give you about 15-20 stacks of the debuff within about 5 seconds if you don't pop AMS.
Priority #1 is to improve her haste rating. That is a HUGE problem. She will notice a MASSIVE difference in survivability and damage once she gets around 30% haste unbuffed. I'm currently at 38% haste unbuffed (865ilvl), and I can tell you that even going from 30% - 38% there is a big difference. Her 17% haste is insanely low. Fix that and she'll be in much better shape. Basically she should prioritize haste over everything else. For example, I have a pair of 855ilvl boots that are haste/crit. I got a pair of 870 boots that are crit/mastery, but due to the amount of haste on the 855 boots, they are better. I suggest gearing for Haste > vers, or haste > crit. Mastery is largely garbage, so I would try to avoid it.
Hope that helps.