I thought you wanted to make an excursion/daytrip from Tokyo. I mean you have already booked your hotel in Tokyo, so you requested the one hour option. And I am still not sure if you can stay somewhere in a hot spring resort outside Tokyo or not.
1. JR pass
When you just plan to stay in Tokyo areas without buying a JR pass, Izu may be out of your options and you should find a hotel in Hakone. And Hakone free pass is a good ticket for someone without JR pass.
2. Mixed/unisex onsen
When I took my Chinese friend to a sushi restaurant, quite a decent one, in Tokyo, it took a bit time before I came to think the vegetable sushi she wanted to eat was "
California Roll".
There might be 3 types of mixed bathing which I can think of.
1. Mixed bathing where you have to share bath with strangers like the Hokkawa
2. Private outdoor bath in the room or you reserve ones in the inn/hotel
When you just want to bathe with your boyfriend or husband, this is a good option. But the room with private outdoor bath is a bit pricy from 15000 to 20000 JPY or more/person.
3. A sort of spa resorts where you have to wear swimming suits (most Japanese people may not take this as mixed bathing, but it is fun)
Hakone Kowakien Yunessun
http://www.yunessun.com/english/
EnoSpa
Fee: JPY2,650
http://www.enospa.jp/access_map/map.html
http://www.fta-shonan.jp/stay/enoshima-e.html
Enoshima is close to Kamakura, a tourst area, and within the excursion area from Tokyo, so it is an idea to bathe there if the one hour request is a must.
The info on Odakyu's freepass
Hakone Freepass
Enoshima-Kamakura Freepass
Odakyu Electric Railway