Just got back from 2 weeks in Japan, so I have a few suggestions
Even if OP is going to LA instead, for anyone else who wants suggestions for Japan:
Tokyo:
http://www.ninjaakasaka.com/top_main.html - Not cheap, but the food is
brilliant. Also, ninja.
Stay in a traditional ryokan for a night. We stayed in
Ryokan Izuyasu which
also had brilliant food.
Kyoto:
http://www.samurai-kembu.jp/ We went to a sumo wrestling tournament, too - surprisingly good fun. (Probably helped that they had a good few girls with barrels strapped to their back wandering around to ensure a constant supply of beer...)
http://www.gnavi.co.jp/en/ is your friend for finding izakaya, which you want to do.
One tip I would give is always have more things lined up for a day, than you think will be needed. That way if you get somewhere and decide 'actually, this isn't as amazing as I thought', you've got an alternative place to go / things to do - rather than just being stuck somewhere with nothing interesting going on. Guess that applies outside of Japan, too