Tokyo with Teenagers. Where to stay?

We will be in Tokyo for 6 days in March. It will be 2 adults and 2 teenagers. I would love to stay in a fun area of the city, lively with lots to see and do for teens. I would also like to stay at a hotel with good subway/train access. We will be taking the bullet train to Kyoto after Tokyo. I would appreciate your ideas. I will be getting 2 hotel rooms and would like to keep it under $200 per night per room ($400/night total) Thank you!

The Keio Plaza in exciting Shinjuku, Tokyo often has some rooms for under $200. The hotel is good value and attractively located:
http://www.keioplaza.com/index.html

Century Southern Tower and Sunroute Plaza in the same area (Shinjuku) can also be considered.

Century Southern Tower
http://www.southerntower.co.jp/
Sunroute Plaza
http://www.sunroutehotel.jp/plaza-shinjuku/