There are loads of ways to find an office job in Birmingham. As you say job centre plus is a good place to start, although be careful as some of the suggestions would be miles away from Brimingham.
Other places to try would be online job sites like Monster, CV Library, and anything local to the area. You could also write up a lot of letters including your CV and mail them (or hand deliver them) to offices in the area that you would like to work at