I'm not sure Thomas Frank would change jobs. He's probably delighted to be at Brentford. While you're a bigger club, at this moment in time, it is probably more attractive for him to stay put. Carrick will probably want to stay where he is as well. It is a risky proposition to come to Leicester now for a young and upcoming manager. I think if you survive, he might consider it, but if he goes down with you, that would be a big blow to him.
As unpopular as it might be, Rafa might be a decent shot. He has the experience, and I think he would not worry about any fallout from a relegation. He'd also be willing to stay if you go down, and he has a proven track record.
I'd stay away from Steve Bruce.