Brussels native here, and still visiting very frequently...
xxx
Two choices for you, either cheap/nice little hotels, with an extended stay rate (could go down to €50 per night), rates depends on the area. All rates include everything, no "taxes" or "fees"... Just everybody speaks English.
xxx
Be careful about hotels near the main train stations "Midi" or "Nord" (especially "Midi") as some hotels are dumps, and area not too nice at night, unless you speak Arabic or Turkish.
xxx
The other solution, you rent a room privately, you could even ask from a bed and breakfast location, in the BRU airport area called "Zaventem" (the main street there is called "Vilvoordelaan") walking distance from terminal building.
xxx
Plenty of info on the net. Do your homework on google.be and get back to me on PM for advice about area, places to eat and transport. Public transports are excellent, subway, trams and buses. Get a "monthly pass" for unlimited travel in the urban area. See "stib.be" for rates and info. Taxis are rather expensive. Inexpensive and excellent restaurants everywhere, opened late, even a few opened all night.
xxx
Airport is located some 15 km NE of city center, trains scheduled every 30 minutes to train stations "Nord, Central, or Midi", from 05:00 until 23:00.
xxx
Happy contrails