I just solo'ed this and wanted to share my slice of the pie of how this is doable.
I play as a Death Knight, 535 ilvl with tank gear and DPS Helm (legendary meta gem), legendary DPS Cloak and DPS weapon with Rune of the Fallen Crusader. Strength Flask and 275 STR food.
Started by equipping Shadow's Edge, popping cooldowns to add to the DPS, get subjected to 4 breaths. Now here comes the tricky part. As soon as you're subjected to 4 breaths you gain a 5min buff (sorry, don't remember the name) that tells you that Shadow's Edge is ready to consume Sindragosa's soul. That gives you time to kill her with your standard weapon.
When she hits 35% she will no longer breath, but spam Ice Tomb (no effect) and melee hit you for the rest of the fight except for the air phases. This is why you start with fight with Shadow's Edge and don't wait until the last 15%. Be sure to pop cooldowns as often as possible to maximize DPS.
I hit berserk during my trial-and-error tries. She hits for around 40k per hit on plate around 2-3 times per second. If your DPS is low I suggest you spec and gear up for this damage. Kill her within the buff limit and you'll be good to go.
tl;dr version:
- Start with Shadow's Edge equipped
- Get hit by breath 4 times
- Check your buffs, if you have a Weapon Icon buff, you're good
- Switch to main weapon
- Kill her before weapon buff runs out.
- ?
- PROFIT