I am sorry to hear you are worried about Jasper. My name is Rebecca. I am a veterinarian and will do my best to help.
Not all parasites are worms, so using a dewormer will not get rid of every parasite. The first thing I would do is to examine a stool sample under a microscope to look for things like Giardia or Coccidia. Coccidia is a common, one celled parasite that can cause diarrhea in kittens. If he has Coccidia, a course of a medication called Albon should help stop the diarrhea.
Diarrhea in kittens can be caused by many parasites, conditions and infections. Most of the time it does clear up as the kitten gets older.
I did take one kitten from the local shelter because he had such bad diarrhea he kept coming back after being adopted. I tested that kitten for so many diseases, did so many tests. He still had diarrhea after he was a year old! One of the toughest cases I ever had. What finally worked for him was being fed only Royal Canin GI diet. He is still on it, and as long as he only eats that diet, he is fine.
Probably Jasper will not be as bad as my cat. Most kitten diarrheas do get better after a short time. Have a stool sample checked, treat for Coccidia if they are there. If not, try a 5 day course of metronidazole for the diarrhea. Consider trying a different food. That is where I would start if he were my kitten.
Let me know what else I can answer or help with. Rebecca