How about Spa LAQUA? It's located at Suidobashi Station which is few stops away from Shinjuku by JR Sobu Line.
LAQUA has got amusement park and shopping mall attached to Spa LAQUA so you can spend a whole day :)
There are many Sento (public bath, not necessarily using onsen (hot spring) water) in Tokyo. Here are some ideas within Tokyo:
If you have got a day to explore onsen, you should take "Romance Car" express train from Shinjuku and go to Hakone.
Hakone has much better onsen comparing to Tokyo and there are also many beautiful ryokans (Japanese traditional inns) offering beautiful bath and delicious dinner & breakfast.
Refer to this article about ryokan in Hakone: