To start this quest, you need to click on an item on the ground located at 50.7, 20.8 in Bastion. Here's a screenshot showing the item you need to click. You will then get a charge bar which will fill up as you defeat mobs in the quest area. After you have fully charged your bar, return to the golden dot shown on your map and click on Forge-Keeper Mnemis, then click on the button on your screen, which will make you charge Mnemis. After charging Mnemis you'll be finished.
According to comments on this page, it seems that for some people the quest is bugged and they don't get an extra button after clicking on Forge-Keeper Mnemis (after fully charging the bar). It seems that for at least some people, logging out and back in has caused the button to appear - but it's possible that for other people the relog doesn't change the problem. If I learn anything new I will update this post.