I did that to my 10/22 before and waited so long to clean it that the rod pushed a plug of lead into the bore. The way I cleaned it was to plug the bbl and fill with Hoppes for a couple days, come home after work, take a couple swipes at it and get some out, refill and do more tomorrow or Friday. It took a couple weeks to come clean.
Personally, I wouldn't use the JB paste until after it was clean and very sparingly. It prolly has a rough spot in the bore catching lead in that one spot.
"Always place your clothes and weapons where you can find them in the dark"
"Understand that the enemy is not the enemy in his own eyes ;this may offer you an opportunity to make him your friend. If not, you can kill him without hate, and quickly."