Hi there, unfortunately I cannot assist outside of this site as that is against the terms that I work under. Going back to your query, it appears they are claiming that you terminated the contract early without giving sufficient notice. The lack of any defined timelines, or terms in writing will hopefully assist you as it would make it more difficult for them to argue you had breached the agreement between you and them.
Whether a self employed worker is entitled to a notice period will depend on their contract. If there is a termination clause that specifies a notice period on termination, the employer would be expected to give that notice if they wish to end the employment relationship.
However, it is often the case that no written contract exists, or there is no notice clause in it. In such situations, the worker can still expect a 'reasonable' notice period to have their employment terminated. This is because even in the absence of a written contract they will be working under an implied common law contract and to terminate such a contract a reasonable notice period is required. The only exception is if the contract was terminated because of gross misconduct, that is any misconduct serious enough to justify the employment relationship terminating immediately.
What is a reasonable notice period will vary greatly and will depend on the individual circumstances, industry practices, length of employment, frequency of payment, etc. There are far too many variables to consider, which means it is usually impossible to give a precise indication as to what would be reasonable in each case. It is therefore down to the courts to make that decision. The worker can nevertheless raise this issue with the employer and attempt to negotiate a reasonable notice period with them, a period that they will both be happy to accept.