I'm still trying to understand what the Constitution and the oath of office have to do with immigration reform. 
The word "immigration" is nowhere to be found in the Constitution. The Constitution gives Congress the power "To establish an uniform Rule of Naturalization" (Article I, Section 8, Clause iv), and "naturalization" is the manner by which an immigrant already in the country can become a citizen.
That means that in the strictest sense of Constitutional law, being that "The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people" (Tenth Amendment), it could be easily argued that power to regulate immigration belongs to the States, and the power to set the rules and methods of how those immigrants would become citizens belongs to the United States.
That was changed, defined or redefined (pick your own word) by way of a series of SCOTUS decisions which all basically argued that allowing the States to decide who came in and who got the boot, and the Feds who could become a citizen would leave us with a hot mess. So, the Feds took over all phases of immigration.
Eventually, the issue of immigration became a inherent sovereign power of the United States.
Having said all that, while the Constitution charges the Executive with enforcing all laws, the Constitution and case law grants Congress the ability to create or "reform" immigration law at will.
The Oath of Office has absolutely nothing to do with the Congressional power to enact or reform immigration laws, and even less to do with the enforcement of those laws, since Congress lacks the Constitutional power to enforce existing laws.
