Fast fire. You've gotten past the hard part, admitting the mistake. The quicker you let someone like that go, the better for you and your team. The mistake here

Fast fire. You've gotten past the hard part, admitting the mistake. The quicker you let someone like that go, the better for you and your team. The mistake here would be to work on their performance and drag it on. Your team morale will take a big hit. At best you'll see a marginal performance improvement over months. At worst you'll waste months on this employee and have to fire them anyway.