There are a couple of examples of why this will not work.First of all we saw in Chelsea that they changed two times coach this season and they are playing for nothing now in Premier League,one of the worst recent seasons of Chelsea,in fact the worse since the era of Abramovich.We have another case at Bayern Munchen where they changed coach while they were fighting for three trophies and they are already out of 2 now,so these examples show that if this has not worked in teams like Chelsea and Bayern which are much stronger than Leicester,most likely won't work at Leicester either.
They stand absolutely no chance against Manchester City especially after the brilliant victory of City over Bayern in Champions League which takes their moral to another level.
With a new coach it won't have much of an impact at the start there are usually twists to be had for example you mentioned Chelsea and Bayern Munich and of course with Leicester as well even though there is a new coach I'm still skeptical it will finish any better with the rest of the season now even with 8 games remaining to move out of the relegation zone it will probably be better for Leicester City than nothing.
It seems that with the remaining season Leicester will have to face some big clubs like Man City and Liverpool but depending on how they do with other teams if nothing changes at all then it won't help them out of relegation.