in addition to the advice above, I don't know how old your kids are, but they might be more receptive to cleaning up after themselves if they know it's for the cat. I know my niece was very diligent to clean up her toys and clothes when she knew that either the cat could get sick or the cat could ruin something she really likes. It's a new motivation besides "mom said so"
try not to give the kitten the opportunity to chew things he shouldn't. Keep shoes in closets, clothes in hampers, and toys in bins or toy boxes.
In a functioning home this certainly isn't always possible to keep everything perfect, but it might help if the kitten has a reduced chance of stumbling upon inappropriate things to chew on.
Bitter apple worked well for my cat. Keep in mind, he's teething, and to him, anything is fair game (particularly cardboard). My kittens are in a very chewy phase as well, but thankfully they mostly restrict themselves to their toys, as there are so many to choose from. Try to make his toys more accessible than the other things he's chewing on.