Can you now set up the network as before with your 10.x addresses and post the ipconfig /all again and also the same details again of the host when it's on the 10.x address (IP SNM DG).
Also, when you have the 10.x addresses configured can you ping the esxi host from the vsphere client machine?
When you say you are connected to a router, in essence no routing is required for this and the simplest way to connect is with a switch, without routing outside of the LAN (more configuration would probably be required in order to get connection to an ESXi host outside of the LAN working). What ports on your router are the host and vsphere pc plugged in to? What make and model of router is it?