There are two pieces here.
If Congress requires a Department called “The Department of Education” then such a place must exist.
It could be a porta potty in front of the Capitol with a big sign on it—but yeah it must exist.
The appropriations are a separate piece.
Over the course of the full fiscal year (ending September) the President is required to make a reasonable effort to spend appropriated funds (impoundment issue).
However unless Congress has given great detail on exactly how the funds are spent the President has total latitude.
Most Congressional appropriation bills are stunningly vague.
If necessary President Trump could just cut big checks to the states to spend the money.
Meanwhile next month Congress will have to debate a continuing budget resolution. That will be their opportunity to weigh in on this issue.
I remember, long ago, a state supreme court ruled that the legislature was not providing enough funding for the court to do its (state?) Constitutionally required duties. The court ruled that the legislature must provide it with more money.