Yes. A handy rule of thumb states that a JR pass costs roughly the same as an Osaka-Tokyo Shinkansen round-trip. So if you use the pass to cover that, everything else that you do with it is "gravy."

I think your Kyoto itinerary is a fairly "busy" one, so I don't think it gives you the best opportunity to lie back and relax in a hot spring. If you had an extra day in Kyoto, I'd suggest taking a walk in the mountains to the north of the city and stopping at Kurama Onsen for a break during the day (or staying overnight).
If you want a relaxed overnight experience with traditional accommodation and onsen, I would suggest looking at your options on Miyajima first. Speaking for myself, onsen is great after a good hike in the countryside. (For other people, the quintessential onsen experience is different. They might explain why they would prefer an onsen in Kyoto. Nobody is right or wrong). There are a couple of onsen ryokan on Miyajima. Prices are not the cheapest because it's a popular tourist destination, but that is also the case in Kyoto. It should help that you're there on a Tuesday night, which should be cheaper than Friday or Saturday.