Hang on a moment. I've just tried my favourite accommodation website and there are plenty of hotels with rooms available in Milan over that period!
Start here:
http://www.hostelworld.com
(NB: Do NOT be put off by the word 'hostel' in that URL. There are some good hotels and B&Bs, at excellent prices, on that site).
Carry out a basic search for 'Milan', with your dates. Then click on 'Filter' to reduce the list to 'Hotels' and 'B&Bs'. Take care to read the reviews (both on that website on TripAdvisor) and to check the location. However you should be able to find something to meet your needs. (I used that site to find a really lovely little hotel in central Madrid, with air-con, wifi, etc, for just €22 per night!)