With education, the majority of funding comes from state and local governments. That has always been true in the US.
This is a pretty handy site to see what the US spends on various items and at what levels of government.
http://www.usgovernmentspending.com
Focusing mainly on K-12, what would I change (I am assuming this is a "king for a day" scenario
1) Eliminate the Dept of Education as a cabinet level position. I think it has value as an infomation clearing house, providing help with international information and student exchanges, provide online resources into federal government and historical documents for open use (ie digitize university collections or provide funds to have Universities accomplish this).
2) Eliminate the public unions from the hiring/firing process beyond being a representative at arbitration hearings.
3) De-politize education. Obviously a tough one, but as two examples ..w e do not need to be teaching "Intelligent Design" in science classes and we do not need half of World War II to be spent on the Tuskeegee airmen (because it happens to be taught in February -- Black History Month) ... I think the Tuskeegee Airmen should be taught, but there is no way anyone can say that that should be a full half of the time spent on what was the most siginificant period in 20th Century history. There is a giant shit storm every time large states try to order new text books about what should be in them and what should not (these text books end up being used by smaller states). So you end you with a history book written by the Texas education dept, one by the CA education department and everyone else picks. Neither is right, both has blatant factual errors, and out of whack priorities.
4) Have a set of standards and expect them to be followed. Teachers/Unions complain they are forced to teach to tests, but how else do you really measure what students have learned if not for tests and papers. Design tests that determine if students have learned what they should over the course of the year. This should be large part of teachers pay and determine if they keep their jobs (see #2)
5) Recognize some students are not in school to learn, not every kid can be saved, some kids are just bad kids (perhaps bad parents, perhaps something in born), in any case for schools where gangs and bullies run rampant, those "bad" students should undergo the following a) separation from kids who actually want to learn b) realize they are probably not going to college and put them in trade schools to learn some skills where they have a change to at least have a job skill c) at the same time invite members of the community into the schools to try to mentor them.
6) Do not allow parents to treat schools as baby sitting services, expect them to help their kids do homework and allow schools top reasonably discipline out of control kids without fear of lawsuits. Parents and teachers should meet monthly (obviously cannot enforce this, but it should be a goal). There is the cartoon (cannot find it) where in 1980 a parent is yelling at their kid for getting an F then the next panel, the parent (in 2005) is yelling at the teacher because their kid got an F. Schools should not be afraid to hold kids back if need be.
7) Allow/encourage some self directed (possibly online learning). Let kids pick topics of interest (even if it is hockey or Star Wars) and write papers/reasearch it. Grade not so much on content, but on reasearch and analytical skills.
8) Beyond "bad" kids, some just are not college material and have interests in mechanical, construction, etc fields. At the latter HS grades, establish trade programs and end the "no going to college shame"
9) More at the college level, but some HS as well. Expand the "non for majors" programs. Teach science and math for people that will not be science or math majors, but geared towards practical uses (like Health Science and diet, math on computers, etc). Things people will use in every day life. The same with history, it is nice to have a basic chain of events in World/US history, but having a student to be able to reasonably compare events or string together some logical courses of events is more useful.
10) Bring back a version of Home Economics, Economics, and Civics as required classes in HS. Students should understand how the government works (in design and practice), at least some basics of supply and demand and basic economic theories, and some practical infomation on things like 401Ks, medical insurance, writing resumes, going on interviews, balancing a checkbook, buying a home versus renting, how doing stupid stuff on Facebook/Social media can come back to haunt you, etc. I am shocked this is not mandatory everywhere. School needs to have some basis in the real world.
11) If schools keep failing, give vouchers and let kids move to private or public schools that are not failing. It does not do a Freshman any good if a school is on a 5 year plan to fix itself
That is 11 I can think of quickly, I guess someone can feel free to explain why the current system is good and why we should spend more money on it.