Both my husband and I have had to let people go who work for our small businesses and who work in our personal lives. We have done this successfully and, unfortunately, sometimes not so successfully. I would say that the toughest thing to do is to have to let somebody go who works in your home. It is much tougher to remain objective when somebody is working alongside you or with your children and, for this reason, I would highly recommend keeping a running list of concerns and celebrations so as you can remain more objective. It is important to address niggles and concerns early, in an informal manner, to allow people to correct their behaviour to meet your needs.
It is also very important to give people the chance to air their concerns and for you to be willing to change your own behaviour to accommodate others. It is not easy having people work in your home or in the business you have lovingly created. However, it is also not easy to work in some body's home. I have found the most successful relationships to be where you have a dedicated time each week to catch up over a coffee about the nuts and bolts of yours and their week and working relationship, in addition to the scheduled meetings you have about the actual work done or your child's progress. Write your expectations down clearly and record what they can expect from you so as all along you can address quickly any struggles which arise.
It is very hard to let somebody go who you have developed a relationship with, but you must remember that this about your child's progress. Be honest and upfront and try not to surprise anybody with what you say to them! Nobody should be shocked that they are being dismissed if you have cultivated a transparent relationship. Always try to get something in writing reiterating what you have said verbally to your staff member within 48 hours of having the conversation.My biggest recommendation would be to be gentle in your actions and in your words at all times and when you feel you should apologise, do so quickly!