Upvote:2
Maybe Yumoto Onsen? It's a bit more than an hour by bus from Nikko. For an Onsen in a Ryokan, check Kinugawa Onsen, which is around 30 minutes by train from Nikko.
Upvote:3
The town of Nikko itself, where the Toshogu shrine is located, is not in a volcanic area and thus does not have any onsen hot springs. You have three choices in the vicinity:
All three are major tourist areas and have accommodation running the gamut from Western-style hotels to fancy ryokan to simple B&B/minshukus. I don't believe there are any free baths, but many ryokan/hotels open their door to visitors during the day for a small fee (around Β₯500 is common).