Sounds like laptop number 2 has a bad connection between the base and screen-honestly, not worth dealing with if it's old.
I would shutdown and unplug the modem, router, and all your home computers for at least 30 seconds...Turn everything back on individually starting with the modem, wait 60 seconds, then the router, wait again and start computer. This reset may fix things. Sometimes routers can need to be reset when not used for a while-are any of your computers hooking up with it? If not-you may need to reinstall the router and reconfigure the network you built at home.