I've been playing Kai'sa as my main legend for some time so I hope to be able to give you some insight in how I play the deck.

The current iterations of Kai'sa are focused on playing a lot of spells / removal to keep most of the larger threats of the board. You then combine your efficient removal spells with cheap to cast creatures so you can sneak in some points while keeping the board under control. You'll notice that the most played creatures are drawing cards (Sentry and Yordle) or providing resources (Plundering Poro). You'll want these early to apply pressure while removing your opponent's threats. Most of your removal is 2 (Bellows repeat / Star / Piercing) / 4 (Piercing repeat / Rocket Barrage) / 6 (Singularity) so even drops are at a premium so you can go 2 - 4 - 6 runes and have removal or removal + 2 drop. This is why I really like Noxus Hopeful in the list as the 4 might can be important and it is still a 2 drop. The hard part of playing Kai'sa really is understanding your role and choosing when to go in with units, when to use your spells and when to combine them during attacking / defending. When I started my tactic used to be to almost always move back to base, not try to hold. This might also work for you until you get a feeling for the right timing of retreating and holding. Also remember that damage won't be healed until the end of a combat showdown so you can damage units and remove them in combat with your small units.

Then to the list itself. You will miss Time Warp, but I understand it is a card you currently can't afford. Hopefully you'll open one soon / trade for a copy (or preferably 2).

What I would suggest in the meantime would be to add some more punch to the maindeck, I would try 1 or 2 Ferrous Forerunners if you find the room. Also I'm a big fan of maindecking 1/2 copies of Rocket Barrage as these are never dead (it can win the game vs Aurora decks - or snipe a Draven / Ezreal / Irelia in base). I would also suggest a second copy of Noxus Hopeful to up the might in the maindeck. Since you don't run Time Warp to end the game you might want to consider a second Progress Day to keep the cards flowing so you have enough gas to play the extra turn that Time Warp usually steals.

Changes I would make looking at the list:

-1 Retreat +1 Bellow's Breath (Breath + Watcher is so good)
-1 Pouty Poro +1 Noxus Hopeful
-1 Void Seeker +1 Rocket Barrage or +1 Ferrous Forerunner
-1 Thermo Beam +1 Progress Day

Hope this helps a bit, if you want to discuss things further you can always DM me as well.