People have mentioned many nice places, but many are above the rate you requested. As I re-read your question, one perfect fit is the Hotel Le Cep where we have stayed 3 or 4 times. It is very historic and located inside the ancient ramparts of the old city. Some rooms are more than you stated but several fit your price range.

I also second La Lucarne aux Chouettes, but it is rather remote and we stay there on the way going or coming to Paris as a charming stopover. The restaurant is it's raison d'etre.

I also stayed at Hotel Le Cep and found it very charming and quite convenient. The town of Beaune is delightful, and wished I stayed longer than the 2 nights.