Yes firing someone can be difficult, but the mindset shift for you is this actually has very little to do with you. This is not an arbitrary decision at all. This person is not working out for specific and clear reasons and part of your job is ensuring the organization is being run properly. Feeling guilty is when you believe you have done something wrong or acted with malice in some way. That simply is not the case, as you’ve described it. Certainly, you can show empathy towards this person when you have the conversation, but the mind shift is mentally understanding the difference between feeling guilt (you have not done anything ‘wrong’) and empathy (you care about this person’s well being).

Good luck!