Sometimes you can remove a stuck cap by inserting a paring or pocket knife blade under the rim and gently tapping and twisting until it pops off. Be careful about getting too vigorous with a piece of cheap plastic!
Corrosion/hard water scale has built up between the slotted shaft and the handle .... "cementing" it in place.
A faucet handle puller may help depending on your design. This video shows how for a 2 faucet sink setup...but your 1 faucet handle should be similar. The puller will ONLY work though if you can access the end of the slotted shaft to get leverage from. Can you remove an end cap from the handle to access the shaft? ...
If the puller will not work ....
You could try penetrating oil (if you can get the oil to the corrosion) ... or if your handle is metal, cover it with a wash cloth to prevent marring and tap with a brad hammer. Sometimes that breaks the corrosion bond.
You could also try heat/cold/heat/cold by heating with a hair dryer followed by a bag of ice water and then repeat. The temperature cycling might help.
In a real pinch ... but you have to VERY careful ... use a pry bar between the escutcheon plate and the handle...go slow .... use cloth protection to prevent marring ... move the pry bar around and rock it....say at 12, 3, 6, 9 o'clock positions around the handle
AND .... when you are successful ... when you reinstall ... put some silicon plumbing grease on the shaft to prevent future corrosion!
the right one. Get a set. Try different sizes till you find one that fits snugly. No wiggle. There are american sizes and metric sizes. Get one of each (probably $20 at home depot) Northern Tool and Equipment way cheaper. Ebay, sometimes cheaper still. Or go to Home Depot or Lowes with a photo of your kitchen faucet and see if they sell one that looks like it, and see what it uses...
The handle needs to return to the up position with out binding (sticking) otherwise the chain between handle and flapper may be taught and when closed causing the flapper to leak. If handle is binding, loosen large plastic nut, on inside of tank that holds handle to tank, by 1/8 turns until it returns to up position freely.
If handle not sticking, check that chain is not taught when flapper is closed and handle up. Even the slightest tension on the flapper may cause leaking. Adjust chain length so it is just loose. Make the chain too long and it may get caught in the flapper. It's adjust by trial and error until right.