Don't spend months torturing an employee that you know is very unlikely to work out. Fire them quickly and compensate them for the situation you find yourselves in.
New hires should be be given 3 months of severance. Old hires should be given 6-18 months, depending on how long they've worked at the company.
Take all that time, energy, and money you were going to spend on the Performance Improvement Plan and hand it to the employee on their way out. They will thank you for it.
It's not just a gentler and kinder way to behave, it's also saves money and improves productivity.