FWIW I very much enjoyed the Oedo Onsen Monogatari which is on Odaiba and not that far from Shinagawa. Yes, this one might be a little too "kitschy" and not like other onsen elsewhere but it's fairly inexpensive, is open practically 24/7 and a offers great way to relax. Some English is spoken by the attendants as well so that could be one way to soak away a few hours.

These ideas are of course only likely to help if OP takes that step and ventures out to experience all the city has to offer which I hope they do as I fully & completely enjoyed my first visit in March.