Seek attention. No antibiotic or amount of soaking is going to take care of this for any period of time, you need to have the toe numbed and the offending nail particle removed. Otherwise, you will continue to have this infection.
See Podiatrist. See your podiatrist. A procedure called a partial nail avulsion with a matrixectomy can be performed where the offending section of nail is removed and a chemical called phenol can be used to eradicate the small section of progenitor cells the the nail grows from so that section of nail never grows in again and you never have a ingrown toenail again in that area.
Begin soak, bandage. Begin soaking in a vinegar/water (1:4 ratio) solution to calm the inflammation and to help soften the nail and nail fold. Apply direct compression on the site that bleeding. Once controlled, apply a bandaid over the top of the toe (not around the toe) to cover. Make an appointment with a podiatrist to have the toenail evaluated and to get options for long-term treatment.