Have to agree with the other posters OP, spending 10 days in the same city will be a bit boring. (For the second half of those 10 days anyway.) I thought this as soon as I read your first post. 4-5 days would probably be enough. 5 maximum.
I have never been to Japan, but I know several people who have - including my niece. She went with her partner last year, and they visited loads of different places, all over the country, as pps have said, getting trains, and staying in hostels and hotels along the way. They went for a month, and had a wonderful time.
Not sure it's what you want though is it, staying in hostels? You could try and book hotels in advance, or see if a travel agent can tailor a trip for you that includes 3 or 4 day trips to other places. It will be a bit boring just staying in Tokyo for 10 days.
I love visiting London, and there is lots to see, but DH and I went for 5 nights/6 days recently, (stayed in a hotel in the West End,) and we were definitely ready to come home on the 6th day. As much as there is a lot to see in London, we started to get a bit tired of it by day 5.
Also, we (me, DH, and our DC) went to New York (stayed in Manhattan) a few years ago, for 4 nights/5 days, and we managed to get around everything we wanted to see in that time. It was totally enough time for one city IMO.