For some reason hotels in Italy are more expensive than comparable properties in comparable locations (e.g. France, Spain). Unless I definitely want a non-chain property, the first place I check for hotels (at least within Europe) is the Accor website. There is an Ibis in Rome and a Mercure in the centre of Milan both of which are around €100 per room per night.
But if this is your first visit to Italy, I personally don't think it's worth going to Milan. Firenze, Sienna and even Modena are IMHO nicer, and the South is even better. As far as heat is concerned, I wouldn't say Italy is a very hot place...Airconditioning is recommended but not compulsory as would be in places like Dubai and Morocco.