The first part of this mission involves walking, then running, then watching cut scenes. Then, it's just a boss battle. Come stocked with Mika tickets just in case, and if possible, the upgraded "Healing" skill. After taking damage, you can hold LB to turn blood into health, and you'll stay fairly stocked with blood if you keep shooting to a minimum (except in the second phase) and keep on the offensive with melee attacks.
In the first phase, she makes use of several attacks:
1. She will throw cross-shaped red things in the sky that shoot at you. You have a couple seconds to prepare, so just be ready to hold B to block them all. She may also shoot at you directly, which doesn't give quite as much notice in advance. Still, just stand and block these attacks.
2. She will create a large round shield thing that shoots a powerful laser. If possible, move behind her. She will turn around though, so it's likely this will hit you if you can't stay behind her. Attack from behind if you can.
3. She will bring up a wavy red line cage around herself. Move away just slightly, and keep attacking with the sword. You might take minor damage from a couple of hits, but the lines go away when that happens, making it easier to attack.
4. She will cast more red things into the sky that shoot down lasers that sweep the area. What worked best for me was to move backward and forward to avoid them. You can shoot at her to decent effect when she's doing this.
When her health is less than half, she'll bring up a round red shield that will track you and always block you. She will continue doing her other attacks while this shield is up, including the larger shield that shoots a laser, which is a little confusing. Keep doing the appropriate counters or blocks, but slice away at the shield and eventually it will go away, leaving her open to more direct attacks. Shooting with bullet shot (the main gun) also helps whittle it away faster.
Once her health is gone, you'll get the RT + X prompt, which requires one blood rose to be available. It fills quickly with melee, so it's likely you can refill it even if it's empty, in time to do the finisher.
In the second phase, she's a little more predictable.
Get in some slicing attacks when she's on the ground. When she jumps up, prepare to dodge. She will do this twice, then stay up in the sky on a red disc platform she creates. You need to shoot it out from under her. She'll jump to the next, then the next. You can shoot the empty ones first, but it doesn't matter. Ultimately, you have to destroy all three so she comes crashing down.
But while she's up there, though, she'll cast cross-like things that shoot at you. Like before, just stand there blocking with B.
When she's back on the ground, this is your best chance to deal the most damage. Get in close and slice away with X, which becomes more effective the longer you do it. After several seconds, she'll jump again (twice), and end up back on the sky discs you have to shoot. This simply repeats until you've depleted her health.
With the RT + X prompt given, and one blood rose to spare, finish her off.
After the ending cut scenes, the mission stats screen appears. As with all the other story missions, this is when the achievement will unlock.