I know that the majority of US state governments mirror the federal government pretty closely, with some variations (I believe Nevada has a unicameral legislature, etc.), but mainly you have a governor, a state legislature, and a state Supreme Court, and functions as a small constitutional republic. And I know that any new state being admitted requires the approval of congress, and on principle, it's extremely unlikely that any state constituted as a monarchy would be admitted to the unions. But is there law against it?
Yes, there is law against it -- quoting from the Constitution, Article I, Section 9 (Powers Denied Congress):
No Title of Nobility shall be granted by the United States: And no Person holding any Office of Profit or Trust under them, shall, without the Consent of the Congress, accept of any present, Emolument, Office, or Title, of any kind whatever, from any King, Prince, or foreign State.
And Article I, Section 10 (Powers Denied to the States):
No State shall enter into any Treaty, Alliance, or Confederation; grant Letters of Marque and Reprisal; coin Money; emit Bills of Credit; make any Thing but gold and silver Coin a Tender in Payment of Debts; pass any Bill of Attainder, ex post facto Law, or Law impairing the Obligation of Contracts, or grant any Title of Nobility.
Article IV, Section 4:
The United States shall guarantee to every State in this Union a Republican Form of Government, and shall protect each of them against Invasion; and on Application of the Legislature, or of the Executive (when the Legislature cannot be convened) against domestic Violence.
So it's pretty clearly right out for the federal government (Congress) to grant titles of nobility, and also right out for the states to do the same (Article 1 above). And then in article 4, we get the guarantee of a republican (that is, elected) form of government.
Now what about a king with a title that's not appointed to them by Congress or a state? That's also out, according to Article II, Section 1:
The executive Power shall be vested in a President of the United States of America.
He shall hold his Office during the Term of four Years, and, together with the Vice President, chosen for the same Term, be elected, as follows: (etc etc)
So this is pretty much not going to work for our hypothetical monarchical state.