DS1 still fed 3 times a night until he was 2 years old when I had to force the issue (wasn't nice, but I was waiting for him to give up by himself but started to conclude he never would (or rather, he never would in time to suit me!) as like you I was shattered and was suffering backache which never seemed to clear up as I never got the rest.
I'd maybe wait until she's over the worst of her teething then try some soft methods to see if they work (sending in DH to try and settle her, offering her a cup of milk (or water) instead, just cuddling her, etc), failing that if you feel you can't take it anymore, you might have to get 'tough'
(I held my son for 90 minutes while he was distraught at not having his night-time milk, it was horrid but it had to be done, and 3 nights later, he woke for 10 mins, I cuddled him and he went back to sleep and he slept through from night 4 on. He was 2 years old though (so not needing the milk, just wanting the comfort and waking through habit) so different situation to you.
I read this: drjaygordon.com/attachment/sleeppattern.html before I nightweaned DS2, didn't follow the guidance, but his narrative made me feel better :-D