The point of Target Display Mode is that your 2010 iMac would display the screen of your new iMac using the keyboard and mouse of the 2010 iMac to control the new iMac. That is not what I think you are asking. I believe you want to treat that 2010 iMac as a dumb secondary display — in addition to your new iMac, and that dog won't hunt, because it is a computer, not an ordinary monitor.
If you want to expand your Desktop on the new iMac, then purchase a quality secondary display and connect to it with a DisplayPort cable (for maximum resolution). When you do this, and power on that secondary display, you will see an Arrangement offering in the Displays panel. From that, you can configure the iMac to have an extended desktop onto the external display.