In speaking with several KSG admissions staff and former students, I would say that Harvard KSG MPP admissions statistics are the following (disclaimer - purely my opinion since Harvard doesn't publish admission stats):
GRE: V: at least 650 Q: at least 700
Ask yourself is your score high enough rather than how high it needs to be. In general, you have to be in the top 20% of GRE takers for that year to make you competitive without admissions looking twice at your score. Typically that range is at least above 90% of test takers on the verbal (~650 and above) and 70% on the quantitative (~700 and above). I honestly don't know how important the essay score is but I'm sure it factors more in cases where verbal competency is not easily demonstrated in other areas such as your GRE verbal score and your personal statement. I did hear that at least 700 is expected on the quantitative since KSG's core curriculum is math-heavy. They will also look at any math courses and your performance in them in undergrad on your provided transcript with your quantitative score to assess whether they think you will succeed in their program. In addition it helps to have one of your letters of recommendation speak to your math prowess such as a professor.
Undergraduate GPA: at least overall 3.5
Enough said. All schools and majors do differ in their difficulty. It helps to have fared in the top of your undergraduate school.
Career experience: at least 2.5-3 years
It helps to have a more substantive position in your field and can demonstrate some professional accomplishments. In forming a student body, KSG looks at professional diversity to see what kind of experiences you will bring. It helps to have moved up your career ladder or to show some leadership potential in your career trajectory in the time you have spent in the workplace. Put in another way, it helps to have a more senior title and show career commitment. It helps to have a letter of recommendation come from a professional source who can speak to your ability to succeed and what you have accomplished thus far as a young professional. Brownie points if the professional source is someone in a leadership position (e.g. Executive Director, Director, President, Chair, etc etc). There are many more things that can be said but keep in mind the MPP is geared toward young professionals in their mid to late twenties who have gotten their feet wet in their career passion.
My hunch is if you pass muster on the three aforementioned things, admissions will read your statement carefully before looking at anything else (just like how an employer looks at the cover letter before closely looking at the resume). There is no foolproof strategy to this. My advice is to 1) tell a story about yourself that is complementary to and cogently ties things in your CV together and 2) identify and elucidate some x factor about you but always in modest fashion. Make sure at least 2 other people read your statement, especially former or current MPPers. Unless you're an amazing writer, be prepared to rewrite it at least a dozen times. Don't forget to explain why you are pursuing an MPP and why Harvard specifically (do your research on the program). In fact, dedicate 2-3 paragraphs to this which could mean 500 words to your story and 500 to why the program. More importantly, this statement also shows your writing skills which is crucial to Harvard's PAE and the program. There is also that extra essay you can write to explain something about your application that you feel is lacking.
Letters of Recs
I would say two things: the importance of letters of recs should be what they are qualified to say and what they say about you. So do not just look at who to get but also what they can say so they are not saying the same things. Most recommendations will want your guidance anyway in what to say. One should come from an academic source who can speak to your academic skills, specifically anything to do with mathematics. Another from a professional source who can speak to your leadership potential and professional success. The third is up to you but make sure the third is not going to repeat what the other two have said. You want a well-rounded recommendation.
One of the more important things that KSG provides is qualitative learning that occurs outside of the classroom among your peers. If you get into the program, you'll be attending classes with world leaders pursuing different degrees sitting next to you, studying with you, competing against you, etc. Think of the relationships and connections and things you'll learn from each other. Do you have something to contribute to them? You have to ask yourself assuming KSG's admissions rate is ~20% and 80% accepted will be admitted: do I have something four out of five other young professionals do not have? Have others been in my position before and therefore my professional history looks the same? Have I been a leader in ways others have not been? Have I done things others have not done that make my life experiences unique to the program? KSG's applicant pool continues to increase making it more competitive and you know they love that.
At the end of the day ask yourself: Am I proud of my application AND am I qualified for KSG?
Applicants who spend time to make sure every i is dotted and t crossed stands out in the mix. A certain amount of dedication does come across if your application is polished. If you feel insecure about something you said or did to your application, you shouldn't have submitted it. Make sure someone proofreads your entire application once complete, not just your personal statement but CV as well.
For the most part we never know why we do or don't get into things and Harvard is no exception. A lot of applicants don't ask themselves if they are qualified for KSG and apply out of a dream or a whim. It should be a strategic and calculated effort that in some way shows in your application. Do your homework on the program. Admissions will have a historical and general perspective as to whether they think you are a good fit including what you will bring to the student body they are assembling.
You should feel proud of your application and feel qualified for KSG. Then at least if you do not get in, you will feel like you did everything you could possibly do and control, and unfortunately were not selected. If it's any comfort, the MPPs I have worked with and worked for do not care so much about what school you went to but what professional experience you have for the job. Sure the H-bomb is nice to drop, but so are 20 or 30 other top MPP programs that are also more affordable.
My two cents.