People's definition of what constitutes "walking distance" is highly subjective and I've learned from experience that what I consider as such mightn't be for others. Ditto for what is a "matchbox-sized room".

With that out of the way, my only experience of downtown hotel in Milan is limited to the Innside Melia in the GalFa Tower which, with all the best intentions in the world, I can't consider as being "near" the Duomo so, while I really liked the place, I shan't suggest it.

Hyatt has one hotel near Centrale station, which is 2.8km away from the Duomo and thus in the same situation as the GalFa mentioned above. There's another Hyatt in via Grossi, within a stone's throw from the Galleria and thus the Duomo. That could be an option: Luxury 5 star hotel in Milan Near Duomo | Park Hyatt Milan

I once had a meeting at the roof bar of The Gray, which I think is a Hyatt hotel The Gray | Small Luxury Hotels (hyatt.com) and it looked suitably swanky.