Ask the nursing expert
Any suggestions as to how to follow up in an efficient manner after I delegate duties? 
Sit down with the staff member who you plan to designate a task to, explain the task and mutually agree on a time for its completion. Write the expected completion date in your day planner, with the task and assignee’s name. 
Make sure that all instructions are clear before the staff member leaves your office. Make another note in your planner of the halfway-until-completion date. At that time, check with the staff member to see how the task is coming along and ask if they have any questions.
On the expected date of completion,  arrange to meet again to discuss the task and its completion. It is always important to offer support and several “thanks for your great effort” commendations along the way. If the staff member shows signs of procrastination, you may want to set up more frequent follow-up meetings.
How important do you think that mentoring is for a director of nursing?
Your staff look to your leadership for strong coaching and mentoring. You cannot tell your staff to “go out there and do a great job” if you cannot demonstrate what that actually entails. 
If you pass a call light without answering it, the subtle message is that they can do the same. If you use unprofessional language, they will also. If you yell at them at the nurses’ station, they will think that this is appropriate behavior for them, as well.
Once your staff feels that you are as good as your word and someone that they can trust, they will look for you to be a part of their professional training. 
You are a mentor to your nursing department, so think about the lasting impression that you want to give them.