there's holiday Inn at the airport
A Quick Google search shows
Premier Inn
Airbnb there's some rooms for £25 a night
https://www.airbnb.co.uk/s/Birmingha...ss_id=qh1x9x61
There's also a company called j9 accommodation which I think is about 20-30 minute drive away from the airport who offer rooms for £65 a week
J9 Accommodation Rooms Page
there's also ibis hotels right next to arrivals and departures @ BHX
hope this helps