I apologize for prying as I'm actually interested in how Rumble, a champion with mobility and damage mitigation with W and the ability to make use of both Morellonomicon and Liandry's Torment--two great items against Illaoi--had trouble with Illaoi mid to late game.
According to the build paths of your recent match you rushed Liandry's while Illaoi was rushing Death's Dance. This was a bad decision; while Liandry's works well with Rumble's kit, it is absolutely terrible against Death's Dance as the DoT applied gets filtered through the bleed passive, and the natural sustain provided from Illaoi and DD means all that DoT is easy to heal back, especially since the Illaoi had no HP items yet except for Doran's Ring (which I can kinda get, but what). Morellonomicon would have been better to rush, not just for her but Sona and Caitlyn (lifesteal) as well once the laning phase ended.
Also you started out giving Illaoi first blood; this does not mean the lane is lost outright as you did get four kills on her after that, but it is a considerable gold advantage especially since Illaoi can stay in lane if she is healthy enough after the trade. By the end of the game the player had 177 cs to your 119 cs.
Overall, the problem I have with Illaoi is that so much of her damage, sustain, and even income early game via Kleptomancy (which is a lot, I am not denying that at all) is tied to landing her E.
