As a hiring manager, I am always impressed with a hand written note. It shows character and is very professional. Trust me, it does make a difference. Don't do both, just the handwritten note. Good Luck!
Getting a healthcare position while you are in school looks great on a resume. Also, join your local nursing chapter. They should have a student membership. Volunteer work is important. Get involved in as many committees as possible while you are in nursing school. Also, make sure you maintain the highest GPA possible in nursing school. You would want to put any academic achievements on your resume. These things are very important on a resume for a new grad because of the limited experience. Good luck!