Don't believe everything you read in the MSM. There are at least THIRTY MILLION people currently in this country illegally, and MOST of them came right across the border, in effect, "at night." They didn't over-stay their visas or come in as students. They snuck in.
In common usage, "emergency" refers to something that is urgent and acute Right Now, but the relevant statute does not read that way. It is any situation or threat requiring timely action.
Our Exalted President would be well within his Constitutional and statutory powers to declare an "emergency" and proceed with reallocating funds from other worthless shit to this.