Leicester is a Midlands city with a lot more going on than its industrial reputation suggests. Founded as a Roman settlement (Ratae Coritanorum), it has a long layered history visible in everything from a chunk of Roman wall in the city centre to the medieval castle and guildhall. Then in 2012, archaeologists discovered the remains of King Richard III under a car park — the most dramatic archaeological find in Britain in living memory. He's now interred in Leicester Cathedral, and the Richard III Visitor Centre is genuinely excellent.
Beyond the history, Leicester has a famously vibrant South Asian community, particularly along Belgrave Road — the "Golden Mile". The food here — Gujarati vegetarian restaurants, mithai shops, and street snacks — is as good as you'll find anywhere outside India. The city's Diwali celebrations are among the largest outside India itself.
The National Space Centre to the north of the city is one of the best science museums in England, and the New Walk Museum has a remarkable collection of German Expressionist art you wouldn't expect in a regional gallery.
York and Birmingham are both easy day trips by train or bus if you need a broader base for exploring the Midlands.