You should check with AU on the transportation from the Tenleytown Metro stop. Its about a mile from

I assume you are driving in. I would stay near GW where the hotels are less expensive than Georgetown (though still more expensive than most cities in the US). It also has the advantage of a Metro in case you also do any sightseeing.
I would stay closer to GW/Georgetown and take the Metro up to American (Tenleytown). When my mom and I were first looking disneyland good neighbor hotels at places for me to live a few years ago, we stayed at the Melrose Hotel on Penn just west of Washington Circle. It was nice, nothing super fancy, but definitely a good place to stay. Very close to GW and then a cab ride over to Georgetown.
You should check with AU on the transportation from the Tenleytown Metro stop. Its about a mile from the metro stop to the campus. AU might run a shuttle from the metro to the campus. I know that they do that for the law school which is further down Mass Ave. Alternately you can hop on a bus at the metro stoop (N2 or M4) that goes to the campus. You can also take a bus from downtown directly to the campus.