How do I treat a bad infection that resulted in a home-ingrown-toenail removal attempt?

See a physician. If you have an infection from attempting to self treat an ingrown toenail, it is time to see a physician. Infections need to be properly managed and if not can result in some serious problems (even from ingrowns!) that range from sepsis (infection of the blood) to osteomyelitis (infection of the bone) both of which have serious consequences. A foot specialist can also be able to help that toenail!
Pain, redness. Redness and pain along the nail border, sometimes with drainage indicates an ingrown toenail. If your ingrown nail is infected, you'll likely need antibiotics, and possibly a nail procedure. While waiting to see your doctor, don't go after it yourself, just soak the toe 10 min/day in luke warm water and epsom salts, use a small amount of topical antibiotic ointment, and cover with a bandaid.