Pod filter is not good for CV type carburetors, I would recommend finding the OEM airbox on eBay, and a K&N replacement air filter, or a UNI filter if the K&N is too expensive for you
From what you've described, your carburetors are in need of serious service, if the bike won't run with the choke when it warmed up, up pilot jets, and idle mixture circuit is clogged.
If your in need of carburetor refurbishing services, let me know.
Also I'd at least check the valves make sure they are within specifications, if not then adjust.
Valve adjustment is a key part of tuning and dialing in the bike.
Here's my write-up on the subject,
I would highly recommend that you DO NOT REMOVE THE AIR BOX, the carburetors on the bike are CV type and need to box to function properly, buy a K&N that fits the OEM airbox(K&N KA-2508 Kawasaki High Performance Replacement Air Filter by K&N Amazon.com: K&N KA-2508 Kawasaki High Performance Replacement Air Filter: Automotive Amazon.com: K&N KA-2508 Kawasaki High Performance Replacement Air Filter: Automotive ) that's the short version.
Long boring but important answer, CV carbs stands for "constant velocity". The function of the carbs is dependent on having a calm reservoir (airbox) from which to intake air at said constant velocity. Pods draw from the turbulent, unpredictable air swirling past them as you ride, which is anything but constant. Other carbs work fine with pods. CV's = no bueno, save yourself the headaches.
Now onto jetting, first you need to tune the idle mixture screws first, I recommend 3 full turns out from bottom as a good starting point, then fine tune them afterwards. As far as buying a jet kit, I personally don't have one on either my 250, or 500 ninjas. I did got up one size on my main jets, due to they are lean from the factory, and shimmed the main needles with a small washer.