You really only want one DHCP server per network (or multiple ones working in a redundant configuration). If you have more than one on a given network, whichever DHCP server answers first is going to be the one that gets to configure that DHCP client. So if you have two separate configurations you want to "push" via DHCP, you can't do it predictably if both DHCP servers are on the same network - not to mention if the IP ranges overlap, you'll have IP conflicts.
However, in your case, it sounds like you really want two separate networks, one for your use and one for your kids. You can do this with what you have.
One of your routers will be connected to your ISP. This router needs to be assigned the internal LAN IP 192.168.1.1, and hand out DHCP addresses on something like 192.168.1.2 through 192.168.1.50. Call this router Router A.
Connect a cable from one of the LAN ports on router A to the WAN port on router B. Set Router B's WAN IP address to 192.168.1.51, and its internal LAN IP to 192.168.2.1. Tell it to hand out DHCP addresses on something like 192.168.2.2 through 192.168.2.50. Set the DNS settings on this router's DHCP to what you want for your kids.
Router A should work normally. If you have things you need accessible via the Internet, make sure it's connected to Router A and setup your port forwarding on Router A like you would on any normal router.
Router B will "get internet" through Router A.
Router B will be double NATed. This means it's very difficult for machines from the Internet to connect to anything behind router B. You probably want this since it's for your kids.
By segmenting your network like this, you'll have a separate network for you and your kids and can do what you want.