How can I get 2560X1080 resolution on my LG 29UM58 monitor, connected to a Mac Mini?
You'll need a 1080 capable adapter or better. See this thread for some options. 


How can a Mac mini drive 3440 pixels wide display instead of 1920?


I'd get the cable matters 4K adapter over the 1080p one if you can afford the extra $4 cost. 

Keep in mind HDMI spec as well for limits on resolution: Can a MacBook Air support 2560x1600 output via HDMI using a Mini DisplayPort/Thunderbolt to HMDI adaptor?