I had an inflamed ingrown toenail. I got it taken out today. How long would it take for the rest to grow back?

Hmmmm be patience. Few weeks. Out should start to regrow within a week but 4 week plus to fully cover the entire nailbed.
Depends... ...On the procedure. If just the border was taken out and no acid was used to burn the nail root, the nail will most often (about 95% of the time) grow back 1mm/month. If a chemical was used to permanently remove the nail, it should not grow back (very occasionally it does, but this is rare).