Near Sky Tree, the Sumida-ku gymnasium looks like it may meet your requirements. Here’s the access link in English.
http://www.sumidacity-gym.com/lang/honyaku_english/access_e.html
And here’s the training room (Japanese). http://www.sumidacity-gym.com/sp/facility/2f_2.php#floor-2f-2-02
Yes there are, for example Tokyo Metropolitan Gymnasium. One time entry is ¥600 and it has the facilities you seek (and more).
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5 Mar, 2024
4 Mar, 2024
4 Mar, 2024