Besides the resolution being 4k instead of 1920x1200 the refresh rate also dropped to 30hz instead of 60hz. Because my graphics card only supports HDMI 1.4a which can not do 4k at 60hz.
So I thought maybe the refresh rate mismatch between the 2 monitors is causing some strange interaction that is making Windows work hard to compensate. With that in mind I thought I was going to have to buy a new low end graphics card just to get a 2nd Display Port or a HDMI 2.0 port to drive the new monitor.
Then I remembered my on-board video! All I did was go into the BIOS, select the option to enable both the on-board and PCI-E video cards and Windows 10 installed the drivers automatically. Then I just plug the display port cable into the on-board video and set up the new monitor on there.
Now all the input lag seems to be gone. So I am thinking Windows 10 might have an issue with dual monitor setups that do not have matching refresh rates?