ISO 9001 5.5.2. says,
Top management shall appoint a member of the organization's management who, irrespective of other responsibilities, shall have responsibility and authority that includes
a) ensuring that processes needed for the quality management system are established, implemented and maintained,
b) reporting to top management on the performance of the quality management system and any need for improvement, and
c) ensuring the promotion of awareness of customer requirements throughout the organization.
It does not say that there has to be a letter of appointment.
If a Director of the company wants the role that's wonderful, one of the best situations to be in. If it were me I'd write the appointment down in the Quality Manual somewhere, perhaps in the org chart (not because ISO 9001 requires me to; it doesn't, but because I personally like to put the important high-level detail in the quality manual), tell everyone, and get him or her to sponsor all the difficult things there are to do in quality, such as scheduling a management review and getting some corrective actions closed.
Hope this helps,
Pat