You can take action re a breach of contract claim, however this would be a case for monetary relief only and as such cases cost you both time and money to pursue for one week's notice would not be commercially viable.
I employ a number of careworkers in my own home, to provide support for my disabled daughter. Last week one of them wanted to give me 2 months verbal notice. She said she was leaving because of the way I treated her (disrespectfully). I was slightly taken aback, as two days prior she had been telling me that she really enjoyed working here. I said to her, if it is so bad here, leave now. (or something similar) I have just had an email from her saying that because I didn't let her work out her notice, she is entitled to financial compensation. I am confused, as she walked out. What is my legal position?
It sounds like this girl is setting you up! Firstly, do you have a Contract of Employment in place? If you do, check it to see how many weeks notice your careworker needs to give. If you don't have a contract in place the statutory minimum notice period is no notice during the first month then one week for every complete year worked up to a maximum 12 weeks. She hasn't given you notice in writing, therefore it could have been a general discussion? If she's now left on what you said to her, you could just pay her the statutory notice if a contract does not enhance the notice period.