Here are my recommendations:
Nikko is well worth visiting. It is a UNESCO World Heritage Site, and the location of the tomb of Tokugawa Ieyasu, the real character on whom the novel "Shogun" is based. It can be done in a day, but it's best to stay overnight. The shrines are amazing, and the original "three monkeys" carving is there. It is easy to miss, as it is on one of the stables, not a main building, but your clue is that nearby will be monks selling small reproductions of the carving. When I was there they were about $10, at today's exchange rate, they are probably $15 or so. But they do make great gifts for lawyers.
Travel there on the Tobu Line train from Asakusa station in Tokyo. You can get there on Japan Rail, also from Asakusa, but the Tobu trains are much more comfortable.
The inn I recommend there is the Hotori-An:
http://www.turtle-nikko.comThe web site shows both the Turtle Inn and the Hotori-An. The Turtle looks nice but I prefer the Hotori-An, which is their 'annex' (hence the 'An in the name) even though it's about a quarter of a mile away from the main Inn. It's right by the river, and the rooms at the back actually overlook the river. Rates are very reasonable per person. Watch out for that: it is common in Japan to quote and charge per person even when two people occupy one room. They serve a western-style breakfast for about $12. They offer dinner some nights but not every night at the Turtle Inn, and you have to make reservations. We did, and had a wonderful meal: soup, salad, a small fish course, sukiyaki, and a small desert, all for about $25. I have stayed at the Hotori-An twice, and will go back there whenever I get a chance. It has its own little onsen, which is really nice.
I took the bus up from Nikko to Lake Chuzenji and further on to Yumoto Spa, where I hiked back down to Ryuzu Falls, then took the bus back to Nikko. That's a day trip, and you probably will not have time to do it, but it's nice to get out into the countryside. That was in May. In February, Lake Yu at Yumoto Spa was frozen over and there was snow everywhere. And it was very cold. No hiking at that time of year, although snow-shoeing seems popular, as is cross-country skiing.
There is also a botanic garden at Nikko. It is more of a research garden, but still worth a visit.
I have visited Nikko four times now, and would go back again.
I recommend the "round the world in 8 hours' trip to Hakone, where you ride the train, small train, funicular, cable car, cable car, ship, bus and train in a day-long tour through an active volcano caledera. With some great views of Mt Fuji.
Check out these web sites:
http://www.japan-guide.com/e/e2358_008.htmlhttp://www.travbuddy.com/Hakone-Free-Pass-v302383http://www.hakonenavi.jp/english/freepass/index.htmlIf you do an internet search for "Hakone Free Pass" you will find lots of references Stop in at Shinjuku station a few days before to reserve a ticket. They will sell you a ticket which includes round trip to Hakone and unlimited use of all the other modes of transport around the tour. I strongly recommend the "Romance Car" option because it is an express train and is comfortable. The alternative is a local train which makes stops along the way and can be crowded. If the weather looks decent, this is a great way to spend a day. Just make sure you get to the North end of Lake Ashi by about 3:30 pm, as the last boat leaves around 4:00pm, and if you miss it, you have to backtrack.
I never got to Kamakura but I understand that is a good day trip. There's a huge statue of the Buddha there, and it is a popular day trip for the Japanese.
Kyoto has a lot of castles, shrines, gardens, etc. I like the International Hotel across from Nijo Castle. Stay at least two nights.