Wouldn’t there be some kind of Non-Disclosure Agreement signed by all people who worked at a certain level or had access to the White House?
Let's hope so:
I don't see how this could possibly be enforceable except for classified information that's governed by statute.
The President's staff doesn't work for the President - they work for and are paid by the American people - so who would the contract/NDA be with?.
I also don't know who would enforce this, certainly not in a way that would survive a new administration of the other party.