Hotel Danemark was exactly what we needed!
It could get hot without air conditioning so keep that in mind for what time of year you're visiting, but it wasn't bad in early June. The windows also don't have screens, so if you don't want mosquito bites you can't keep the windows open at night, which adds to the heat problem.
The shower is very, very tiny, but it is located in the room, which is convenient. The toilet is in the hall.
The location is about a 25 minute walk or 10 minute drive from Old Nice, and a 20 minute walk or 7 minute drive from the train station. We didn't mind long walks, but people who don't love the idea of spending a lot of money on cabs/ubers or walking significant distances might not love the location. It is very close to the beach though, and walking along the beach to Old Nice was lovely and definitely didn't make the 25 minute walk feel like a chore.
The owners are lovely and always willing to help with whatever you need! The only down side is that they aren't available after 8 PM, and since many restaurants in Nice don't even open for dinner until 7 or 7:30 PM, you'll be on your own once you get home from dinner. But they have a refrigerator of wine, beer, and non-alcoholic beverages that you can take at any time and are very reasonably priced, which was wonderful.
For the prices at Hotel Danemark, we were so impressed with how wonderful it was and would highly recommend it to other young couples!