Children are harder to discipline than adults, because they are more likely to push the boundaries, they are less likely to take responsibility for their actions, and they'll frequently have a parent who thinks the sun shines from their wazoo and their little angel could do no wrong.
As their new instructor you have to let them know in no uncertain terms exactly where the boundaries are and what is acceptable to you. They need to know that there are differences between you and the previous instructor. Do it verbally to the whole class (several times if needed to make sure you inform everyone, repetition is good), then follow it up with an informative news letter which you also expect the parents to read.
My sensei runs a dedicated kids class, and he introduced a home/school progress component into the gradings. In other words, the kids couldn't progress to the next belt unless they had reports from the school and from home that they had been good - this meant that homework was done, bedrooms were consistently kept tidy, allocated tasks at home were done, and general behavior was good. This approach to teaching and grading has seen some excellent results - and the parents love it and support it.
While you do want the kids to behave and show respect in class, you also have to remember that they are kids - so you'll need to be a bit less traditional and a bit more fun. Kids respond to happiness and fun. But if punishment is required, don't punish the individual - use the power of the group and punish the whole group, make the whole class do 10 or 20 press ups. If you've built a good rapport within the group this approach should work well, the peer pressure within the group should help stamp out problems.
If individuals are not bonding well within the group or are consistently badly behaved, then have a talk with the parent/caregiver - you can't have one person spoiling the progress of the class. It's better to pre-emptively drop one student than to eventually have several others drop out due to that one.