If the DLC is not showing up for you in Adventure mode, but you have the latest patch you will need to download the key for DLC from the Xbox Store. Find Remnant II in the store, once you have found it, you will need to go to the Add-ons for this game section of the store page. Then select the Add-ons for this game / Show All box.
#Installing DLC on Xbox
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From there you should see the Awakened King DLC and it should say Owned. Select that.
That will take you to this page, from there you just need to press the INSTALL You Own This button. Once complete, start up Remnant II and follow the instructions here: https://discord.com/channels/439222927876030464/1174060743772078080