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.