I love Hakone, and I go there primarily for the hot springs.
I do agree that winter is the best time to go for onsen, esp in places like Hakone where it gets pretty hot in the summer.
Hakone's got some amazing hot spring places, but some mediocre places as well.

I've probably gone to more onsens in Hakone Yumoto than anywhere else in Hakone, simply because I mostly do day-use onsen visits in Hakone and Yumoto is the easiest part of Hakone to get to from Tokyo.
Ones I've been to range from mediocre to average in terms of onsen quality.

Some of the best onsens are in Gora, Ubako, Sengokuhara areas.
Many of the hotels in that area tend to have their own onsen supply ('gensen'), and I particularly love the milky-white kind ('nigoriyu') that's unique to that area of Hakone and especially if it's purely from the source without being circulated ('kakenagashi').
It's been >2 years since I've actually stayed overnight at a hotel in that area. With so many changes in the area, I can no longer say I'm up to date on which places are good.
Gora Hanaougi is great, but it's also expensive. Hakone Green Plaza at Ubako Onsen has room type called Asian Room (or something like that) with its own en-suite onsen that has nigoriyu. Their communal bath is neither nigoriyu nor kakenagashi, but the Asian Room's is.
Again I haven't stayed overnight at these places in a few years now, so I may be giving you outdated info.

If you love onsen and think you'd enjoy a lot of R&R time, then I think 2 nights is fine. Otherwise 1 night would be totally fine. It does get expensive staying 2 nights. You can do Hakone sightseeing both before and after checkout. When I'm spending a lot of money at an onsen ryokan, I generally stay put from check-in to check-out and thoroughly enjoy the property, onsen and meals.