Bruce agreed a new deal to stay on as Hull manager on Wednesday, ending speculation over his future with his previous deal set to expire at the end of the season.
With one defeat from their last five matches, Hull have eased their relegation concerns and Bruce is now determined to cement the club's place in the top flight.
"We have to build the club the best we can and have something in place so Hull City will always be a Premier League football club," said Bruce.
"We've had a great two or three years, the big thing we need is to carry that on and that has to be my aim, to establish us as a Premier League club.
"We've been talking [about the new deal] for a while now and the big thing for me was to stay in the Premier league, so it has been pushed aside.
"Once I made the decision to stay, I'm delighted to do so.
"I've had terrific support from the owner and from the supporters, who have been fantastic with me."
While Hull's recent form has seen them move five points clear of the drop zone, Leicester are four points adrift of safety and winless in six league outings.
However, manager Nigel Pearson - a former Hull boss - hopes the return of goalkeeper Kasper Schmeichel can help shore up his defence for the season run-in, although he does concede he is now faced with a selection headache, with Ben Hamer and Mark Schwarzer having deputised in the Dane's absence.
"Kasper's back in contention which is good, so three keepers available which is always nice," he said. "It's been good to get Kasper back into full training. He's been out for a while.
"I will make a decision on what is best for the team going forward. Nobody's an automatic selection. It is down to a number of factors.
"They are all very good talkers. Mark Schwarzer's an excellent communicator, they all have different strengths and it's nice to have them all back available and that's the most important thing."
Anthony Knockaert will be missing for Leicester due to personal reasons, while Hull remain without Liam Rosenior (hamstring), Robert Snodgrass (knee), Robbie Brady (calf) and James Chester (shoulder).
Mohamed Diame could return after three months out with a knee injury.
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