For the first appearance, just get into a fight - she'll pop up then disappear.
For the second appearance, you need to have conquered the APARTMENTS and the BRIDGE and (and this is important) have done the night time event on the bridge that reveals the Shopkeep. Once that's done, sleep until you get the Claire event. Then sleep until Lyx pops up a second time. This will give you a LEAD. Follow it to unlock Lyx :)