How can I get 2560X1080 resolution on my LG 29UM58-P monitor, connected to my 13" MacBook Pro?
According to MacTracker, your 13" 2013 MacBook Pro can support:


up to 1920 by 1080 pixels on HDMI;
up to 2560 by 1600 pixels on Thunderbolt.


So you need to swap the connectors used by each display.