You didn't say where you'd like to spend the relaxing time (Kyushu? Hokkaido? Somewhere else?), but here are a few suggestions.
Miyajima - You can spend 2 nights here. Spend one day exploring the island, including the ropeway part way up to Mt. Misen and then hike to the top (it's not crazy strenuous, and the view of the Seto Inland Sea from up high is incredible.) There is a main pedestrian street with row of shops and eateries that is also fun. And, of course, there is Itsukushima Shrine (and the Torii gate) as well as Daisho-in nearby which is especially picturesque when lit up at night (do NOT miss Itsukushima and the Torii when illuminated, too!) On departure day you can swing by Hiroshima to visit the museum and Peace Memorial and enjoy the local Okonomiyaki. Yes, it's a bit of a touristy place, BUT it's touristy for a good reason (it's awesome), and most visitors are day trippers that leave late in the afternoon - after that, the island is (almost) all yours.
Zao - There are a few Onsen towns near the famed Mountain range in Miyagi/Yamagata, and it's only a little over an hour away from Sendai by vehicle (not sure about trains.) The Zao Echo Line, a wonderful road that winds through Zao that is accessible usually from late April to mid-October, has numerous hiking trails along the route, as well as Okama Crater, which is among the most gorgeous crater lakes you will ever see.
Kakunodate (Akita) - This former samurai town is particularly photogenic when the Sakura are blooming, which typically happens in late April. Time is right, and you'll be in for a treat! For your nature fill, Lake Tazawa and Keikoku Gorge aren't too far away.
Kumano Kodo, specifically the Nachi Taisha (Shrine)/Nachi Falls area - Kumano Bettei Nakanoshima is not Amanemu-priced, and has one of the most incredible communal onsen baths in all of Japan, and has views of the ocean. It's a 5 minute ferry from there to the mainland, and from there it's about 20 minutes to Kumano Nachi Taisha. You can walk as little or as much as you so desire. If you're short of time, just walking up the Daimonzaka steps will make you feel like a buddhist pilgrim. The forest is beautiful. If you'd like a longer adventure, you can hike part of the Kumano Kodo network of trails from Nachi Taisha.
Hirosaki area (Aomori) - The Oirase Gorge is incredibly scenic. It'll be VERY green in April/May - and less crowded than during the Koyo (autumn leaves) season. Definitely worth checking out. You can either drive around and stop at certain points, walk the entire trail, or even rent a bicycle. You can also visit Hirosaki Castle, which is one of the handful of "original" (not rebuilt post-Meiji era) castles with its Tenshukaku (keep) intact. The Sannai-Maruyama Jomon Historical Site (recently awarded a UNESCO World Heritage Site designation) is a good way to see how the people in the region lived in prehistoric times.
Unzen (Nagasaki) - If you're up for a bit of a challenge, hiking to the summit of Fugendake is an unforgettable experience. Down at the base is Unzen town, which is littered with some nice Onsen ryokans, including Hanzuiryo and Kyushu Hotel, neither of which will blow your budget but will provide a comfortable stay. The Onsen quality is A+ here, and is somewhat similar to Noboribetsu in Hokkaido.
Yufuin/Kurokawa Onsen - I second freecia's suggestion on both of these places. While Takefue is the #1 ryokan in Japan, in my opinion, there are quite a few midrange options in Kurokawa Onsen that are still very atmospheric and comfortable. Ditto for Yufuin; it's littered with solid accommodations across the entire price range. Kurokawa is about an hour from the Mt. Aso area, which is worth a visit. At Yufuin, you're not far from Beppu (another high quality Onsen town.) And, for your nature fix, you can take the Beppu Ropeway to the top of Mt. Tsurumi, and from there you can go on a not-too-strenuous hike with incredible views all around.
Finally - as freecia mentioned earlier, the majority of ryokans have Western-style rooms, at least as an option. Even rooms with tatami floors are often there for cosmetic purposes; there will likely be a proper sitting area with couches and tables. This is the case with Kumano Bettei Nakanoshima, Kyushu Hotel, Chikusenso, Hanzuiryo and Miyajima Kinsuikan. As we all know, Japan is a rapidly aging society, and its elderly have as much difficulty sitting on the floor for long periods of time as Westerners do
Regarding luxury ryokans, if you forgo the top "special rooms" many of them are often far below Aman-level pricing. You might be pleasantly surprised at the options that might fit within your budget.