How would you get rid of an ingrown toe nail without doing surgery?

Homecare. If there are no signs of infection you can soak the toe in warm water with epsom salt to soften the area and then pack a small piece of cotton under the nail edge. This may lift the nail away from the skin to avoid needing removal of the toenail edge.
Removal... ...Is the only permanent solution. Just the painful border can be removed, or the entire nail if necessary. After removal you can let the border grow out again, or your doctor can use a chemical to stop the nail from regrowing. If you're concerned, don't perform bathroom surgery on yourself. That can lead to an infected ingrown toenail, which could require antibiotics.
Ingrown nail. To get rid of it as you asked , the only way is to remove the offending border. Not the entire nail. An ingrown nail is a deformity of the growth center or matrix just under the skin at the base of the nail. That will not change as if by magic by playing with it. See your podiatrist , get rid of it and you will gain the relief you seek .
Possible solution. I would recommend seeing a doctor for this problem. Many times a surgery is not necessary. You can try putting cotton underneath the nail to lift the nail up. Warm soaks are most beneficial.