If I was going to be working at the Royal and didn't want to be too far out, and wanted somewhere 'naice' I would go for either Clarendon Park, as pp mentioned, Knighton, Stoneygate or Oadby. Clarendon Park is very trendy, I wouldn't mind living there, but the housing is relatively expensive mostly victorian terraced so less chance of a lawn for your DC to play on. It is right by the university, a lot of lecturers live there I believe and of course many students if that bothers you!. There is a big park (Victoria Park) close by and a nice atmosphere. Knighton, a little further out along the London Road, is typically interwar semis, so more space, but still fairly expensive. The primary school there (Overdale) used to have a very good reputation, unsure if that's still the case. Neighbouring Stoneygate is sort of the premier area to live in Leicester, a lot of big detacheds, but also many flats, and all quite expensive. Still if you were coming from somewhere like the South East you would probably still think it was cheap as chips!
Less posh, but perfectly fine are Wigston (used to live there myself), and possibly Aylestone or West Knighton.
Then there are the villages, like Enderby, Narborough etc which are fine (although apparently Broughton Astley is to be avoided as it's cliquey according to a recent thread!). Posher villages are Kibworth, Great Glen. The villages will all mean a longer drive, obviously.