CrackforBrumak's solution is spot on, but in regards to strats you really do not have to do much if Mildred is there.
The Maneater known as Mildred will take 99% of Quelaag's attention - all I did was hang back and shoot her with arrows. Mildred is a tank, so most of Quelaag's attacks, although deadly to you, on your first playthrough will be nothing to your friendly NPC.
Let Mildred take the lead, and just keep your distance. If you have magic, either from the sorcerer tree or the miracle tree, then you can get this done faster. Pryomancy does not work as well (nor should it - she is fire!
), and arrows will take a bit longer, but they still work.