We live in a world where an increasing number of things that are supposed to be done by the Congress are done by the executive. Treaties are supposed to be approved by the President with the ratification of two thirds of the Senate, but the President makes many agreements on his authority based on a variety of claims. Legislation is supposed to be passed by Congress but agencies do most of the legislating through rulemaking. And wrongdoing by executive officials is generally reviewed by executive officials rather than by congressional impeachment. Some of these actions may be constitutional under the original meaning, but some are not.
Another way that the executive takes action that is supposed to be shared with Congress is through appointments. The Constitution provides that appointments are to be made by the President with the advice and consent of the Senate, but the President often makes appointments on his own. Sometimes this was done through a broad understanding of the Recess Appointments Clause (which a majority fo the Supreme Court accepted, but see here for what I regard as the better view).
At other times, the President makes unilateral appointments of the White House staff on the ground that these staff members – however important their positions seem to be – do not constitute officers of the United States under the Appointments Clause. The reason is that these staff members are said merely to give advice, to engage in nonbinding conversations with officials, or to convey messages from the president to officials – actions that would not involve the exercise of substantial authority and therefore would not implicate the Appointments Clause under the Supreme Court case of Buckley v. Valeo. The public policy problem is that the number of these White House staff members is growing and therefore the power of the federal government is increasingly exercised by people who have not secured the consent of the Senate. Again, some of these actions concerning appointments may be constitutional under the original meaning and some may be unconstitutional.
What can be done about this? The simplest way to address it would be to pass a limit on appropriations to pay members of the White House staff. This would be clearly constitutional and would limit the staff. But the President would be likely to resist such legislation with a veto.
Another possible law that might be employed would be to limit the actions of the staff. While the White House staff can communicate messages from the President to lower officials without being subject to the Appointments Clause, the White House staff cannot give orders to these officials, without being subject to the Appointments Clause. Thus, Congress might prohibit any staff member from giving orders to such officials (and subject a violation of the prohibition to a civil or criminal sanction). Perhaps Congress could go further. They might believe that White House staff sometimes uses uncertainty about whether the President has issued an order as a means of exercising control over officials. Therefore, they might require that the staff member make explicit, in each conversation, whether the advice/order is from the President, and make it a a civil or criminal violation to fail to do so (or to misrepresent the nature of the communication). While the President might resist this law, it would probably be constitutional and would place limits on the power of the White House staff.