Sharks, who won the League with a dramatic 87-85 win over Manchester Giants on Friday, meet eighth-placed Leopards in the best-of-three series, with the winners playing at Wembley on the first weekend of May.
"Last season we were the League champions but lost in the quarter-finals to Manchester. Giants did it to us then, now we are going to have to do the same to Sheffield," Mims said.
Although Leopards lost their first two League meetings against Sheffield and were beaten 67-65 in the Cup final in January, they beat the Sharks 84-77 in the League a month ago.
Giants have to lift themselves after the defeat against Sheffield, but must first decide whether to appeal against the two-game ban on coach Nick Nurse, which could keep him out of the quarter-final series with Birmingham Bullets.
Bullets, last season's play-off champions, were the victims of Saturday's surprise result when Leicester City Riders won 87-85 at Derby Storm. Four late points from Leicester's Gene Waldron forced a three-way tie which resulted in Newcastle Eagles finishing fifth, Derby sixth and Bullets seventh.
Quarter-finals: Sheffield v Greater London; Manchester v Birmingham, Thames Valley v Derby, London v Newcastle.Reuse content