But, you'd have to spend some money (probably <$100).

The switch doesn't do anything but expand the network; it doesn't hand out addresses, doesn't require static addresses or dynamic addresses. If your router is handing out dynamic (DHCP) addresses, then your kids' computer would get a dynamic address--unless you explicitly give it a static address.

It would solve your problem.

And it's way, way less complicated than I just made it sound.

I am the Doctor, and THIS... is my SPOON!