What kind of facility/agency do you work for? Presumably you answer to someone, notify that person. I have never heard of having to notify the state of resignation... unless you work for a state-run agency, but even then I assume you'd know who your supervisor is.
As Director of Nursing, you need to notify your state Department of Health, Home Health Division or whomever licenses your facility of resignation. Additionally, should be notifying DOH who the Acting DON will be. Depending on regulations, their resume may be needed.
Please check your personnel policy manuals. Most Directors level positions require 4 week resignation notice to permit transfer of powers and responsibilities.
It is the Administrator's responsibility to notify the State agency of changes in administrative personnel. There is no Federal regulation that specifically requires this notification. However, your State HHA regs may. Enjoy your new career!