You might try to go private; you don't have anyone screening, and it will be alot more time for you, but it also would be less expensive.
I was an au pair for a year in Germany. While my family was iffy (I worked waaaaay more than any other au pair I knew), I *loved* the children I cared for, particularly the little boy. I didn't earn any significant amount of money, but that wasn't the point; I wanted to learn the language and culture, they needed live in childcare.
While I went through an agency, I knew of several au pairs who did not. Their own families were friends with a friend of a friend of the family they were working for, that sort of thing. Basically the families offered to pay for room and board, transportation (in the form of supplying a bus pass and/or a bike) and a small monthly stipend, usually a couple hundred dollars. But they also weren't expected to work 40 hours a week; usually between 25-32 or so. Light housekeeping and yard work were often included in the bargin. The au pair was expected to pay for her transportation to and from her country. If you work privately with someone who isn't looking to earn money, but rather really looking to experience the culture and learn the language, privately is probably cheaper than an agency.
The au pair needs at least her own room, preferably with her own bathroom and outdoor entrance/exit. If you aren't going to supply a private entrance, you cannot wig out about when the au pair comes and goes when she is off duty. My family did this (even though I had my own private entrance), telling me they were responsible for me and I couldn't stay out all hours of the night. I told them I had not been under my parents' roof or responsibility for over three years (I was 21 at the time, and had been out of the house, supporting myself since I turned 18), and I didn't sign up for parents when I signed on. They weren't used to such independent au pairs (they had previously had somewhat younger au pairs), but once we reached that understanding, things were better.
I think you should look around. If you are in the army, you should have the opportunity to contact families in another country who have late teenage daughters (or sons) who might be interested in doing a private deal.
I don't know how it works with taxes, health insurance, etc. I had a rider on my own health insurance that covered me for that year. They didn't pay taxes on me, but that is because there is a law (so they told me) that au pair work is considered a cultural exchange, and the pocket money is not taxable. I didn't get a visa, and no one ever asked me for any paperwork. I imagine here it will be much more difficult to get the visa, green card, whatever.
Good luck! I hope you can make it work!
Lori