Invent something super funny involving you being a secret agent who needs to infiltrate Japan to stop an international plot involving Yakuzas, Ninjas, and an old Samurai Brotherhood. Make it so it is obvious that you are writing this "for fun" (ie. you are not a delusional paranoiac), but show that you are well cultivated on the whereabouts of the country and what you are talking about. Show you are motivated by demonstrating knowledge. Your writing skill (provided you have some) will have to keep them reading through the whole text, even if it looks "not serious" at first. Hook them, make them read you.

End with something like, "I hope that you will understand the importance of my request, and why it is really necessary for me to be allowed to join the program; thank you by advance, and wish me good luck!"

Make them laugh, I say. The people who will read you are human beings. If they see that you know your stuff and that you are sympathetic to them, there is a high chance they will accept you.

If they don't you'll still have written a nice text. Which by the way I encourage you to post here.