Danny Wilson has completed a triple signing following United's tour of Malta.
On returning to the UK, the boss concluded the finishing touches to all three deals over the weekend.
Hill, who had a successful loan spell at Bramall Lane at the end of the last season, pens a two-year deal following his release from Blackpool.
Versatile defender Tony McMahon, formerly of Middlesbrough, has also agreed a two-year stay at the Lane.
Meanwhile, 24-year-old Miller arrives on a two-year contract, with the option of a third.
All three players could make their debuts against Rotherham United on Tuesday night at the New York Stadium. The game, which is all-ticket, is sold out for Blades supporters.
More reaction to the triple signing to follow on sufc.co.uk and BladesPlayer.tv