E.A.
I have one more postion for you to try.
Put the baby on his back on the bed or floor. Get down on all fours over him and nurse him like that, with your breast hanging down. Often when the point-their-chin position doesn't work, this will.
And you have to be very diligent about the warm compresses, it might take another day or two, keep them on as long as possible. Keep massaging it as long as you can stand it. And at the first sign of flu-like symptoms, get to bed, and stay there and nurse as much as you can until it feels better. Yes, the cabbage leaves work too, and lots and lots of fluids. If you are feeling ill and don't get better after 24hrs of complete rest, go see the doctor, you have mastitis and at that point definitely need antibiotics..
Unfortunately, you will probably have to take a day or two off from work. This might be a good time to explain to them that you have to pump at least every four hours, because if you don't this is what happenes.
I hope you are feeling better soon.