No, for two reasons.
The Ethernet splitter just allows two Ethernet links to run over a single cable. You don't have two Ethernet links, so the splitter doesn't help you.
But also, you can't build out from a client connection. When you make a client connection to an access point, you only have one hardware address. The access point (which is what your router is acting as) will register the first hardware address it sees as the other end of the client connection, and the other device won't work. The WiFi specification requires this behavior of the access point you are connecting to. So a switch won't help you either, sorry.
You could get another mini router. You could get a wireless router that is specifically designed to bridge multiple devices to a client connection. You could connect some device that supports Internet sharing to the mini router and connect the other device to it.