Hey I did a quick search using the requirements you stated:
I found a place that is 136,000 JPY/month + 5,000 community fee, No reikin ("thank you money"), 2-months deposit, 2LDK, located in Hiroo (Minato-ku), 10 mins from Hiroo station Hibiya Line and 20 minutes to Nihonbashi changing trains once at Higashi Ginza to the Asakusa Line).
The building itself is relatively old (built 1977 so if that bothers you, there should be other places in the vicinity (might be newer but more expensive).
I like the area it is located, called Moto-Azabu, which has great proximity to Roppongi, Ebisu, close to an international school (if you got/planning kids). Ebisu Station on JR Line is only one stop away for convenience to Shibuya, Shinjuku, etc.
The area is also located close to many foreign embassies and you can bet it's a safe area (not that Japan is that dangerous to begin with).
I used to work in Minato-ku and loved the area with its bars, restaurants and locals.
I write a little bit on apartments in Japan (
http://apartment-japan.com)in general but if you need more info, just private message me on YIM (markleyjeffd),facebook, twitter (jefmarkley).
Good luck with your search.