I agree that 3 weeks in Tokyo is a lot. If you want to stick with this plan there are many gaijin houses where you can get a room cheap for 3 weeks. See
Guesthouses or Gaijin Houses.
February is slow season and many ryokans/minchukus have special deals. This is also an ideal time to visit hot springs (onsen). There is no better way to connect with Japan in the winter. See
Japanese Guest Houses - Hot Spring Map of Japan and List of Hot Springs in Japan and
Japanese hot springs (onsen) for more ideas.