Hi Laurent
I don't know much about North Tokyo except as you said, it is cheaper. Hopefully someone else can help you with those areas specifically.
I'll tell you a bit more about several apartments I had and the cost because you might be pleasantly surprised. My husband and I were not on an expat package (we were local hire) so we paid 100% of our rent ourselves. There are 2 places we lived with our son, both in very nice areas in Tokyo.
Yutenji - 2SLK - 150,000yen - approx 65m2
This was a ground floor apartment approximately 12 minutes walk from Yutenji station. Contained kitchen, small lounge room, laundry, bath, toilet, massive tatami room which at various times we used as a bedroom with the cot in their too, and later a lounge/play room, another tiny tatami room which we later put the baby's cot in, and finally an average size room. There was also a small outdoor area to hang clothes with a view of the playground (and car park) and our very own persimmon tree. The apartment was from the 80s I believe but renovated.
Musashi-Koyama - 3LDK - 220,000yen - 100m2
This was definitely my favorite. A spacious open plan lounge/dining/kitchen plus three bedrooms, one of them quite large, the other two average sized. We ended up keeping the computer in one plus guests got to sleep there. This apartment was also from the early 80s, a mammoth building with many apartments that was quite unattractive to look at (made me think of a prison) but with the caretaker working daily plus several cleaning staff, it was kept impeccably clean. The apartment itself was renovated and looked lovely with shiny floorboards, a washlet toilet, a fantastic bathroom with large bath (not small Japanese style) which was programmable (fill at certain temperature at set time, keep warm feature, etc). Excellent storage throughout the apartment. Two reverse air-conditioners, a built in combined oven/microwave (it's not that common to get a proper oven in Japan), kitchen also came with stove/grill. Long but thin balcony with city view (we were on 7th floor) to hang clothes on. Only downside was low ceilings but we quickly got used to it. All in all, a fantastic spacious apartment you could easily raise 2-3 children in only 6 minutes walk from the station and an impressive shopping area.
If your company is paying half your rent, you may want to consider living in a more convenient area in an older but renovated apartment. The areas I mentioned in my previous post won't fit the bill as they are too expensive but other areas in should not be difficult to find something (for example on the Tokyu Meguro line or further along the Tokyu Toyoko line). To do this you will most likely need to sign a 2 year lease through a regular Japanese real estate agency, i.e. not the ones that deal with expats on generous packages. They will most likely want the 2 months deposit (most of which we got back on leaving), 2 months key money and 1 month agency fee. Feel free to negotiate on both the rent and key money. I did and I think I only paid one month key money and rent was reduced a little. If you don't speak Japanese it will be easier to get help from your office. That said, my Japanese was very basic and my partners only a little more and we never had a problem. There will be the occasional agent that doesn't want to deal with you but for the most part, the agents we've talked to have been really helpful.
Best thing to do is take a walk around an area and look at the ads outside the agents windows. Agents in Japan will also have access to apartments represented by other agent so what you see on the window is only a fraction of what they can show you.
Sorry about the rambling post. I'm writing it while my son is playing a very loud computer game!
Gamze