mini mid 2010 does not operate with ultra wide screen LG 2560x1080

Go to Apple > System Preferences > Displays > hold the Alt (option) key and click on the Scaled button.

If the display does not work or you can not select a suitable resolution, then please answer the following ?'s.

1. What model is the LG display and what input connections does it have?
2. What cable and connection are you using between the Mac Mini and display?

FWIW the Mac Mini's Mini DisplayPort output supports the higher resolutions up to 2560x1600, while the HDMI port only supports up to 1920x1200. If you are using a direct HDMI to HDMI connection, then you may need to use a "Mini DisplayPort to DVI" or "Mini DisplayPort to DisplayPort" adapter from the Mac Mini's Mini DisplayPort output port.

see the "Graphics and Video Support" spec's at > https://support.apple.com/kb/SP585?locale=en_US

What happens when you hold the Alt (option) key and select Scaled?

Sadly your Mid 2010 Mac Mini only supports 1920x1600 from the HDMI port and that Display is not a very good match for the older Mac Mini's.

What DisplayPort adapter and cable? Some only support up to 1920x1600.