Hi there and welcome to TFF!
I've never heard of JAD (it might be just fine, most of them are) but if you want to be completely sure then pick up some Seachem Prime - its what lots of us use and its so concentrated that it works out to be the cheapest, while also having a great record for working correctly at other features also.
Now of course, it might have nothing to do with your conditioner. You say you've been fishless cycling for a month but you have not had a thread here with a daily log of your testing results so that the members can help. This is the best way to avoid problems in a fishless cycle. There are various problems that the members can spot and help with if there is a visible log. The real story is not about the results on a given day, either, often its important to see the trend over a number of days.
How is your pH? Fishless cycling creates nitrate(NO3) as an end product and NO3 that is in solution will dissociate and about 7% of its concentration will be in the form of nitric acid, causing the pH to fall as the water becomes more acidic over time. If the pH drops down past the 6.2 mark fairly rapidly, the bacteria will stall out and once the pH drops to 6 and below they will usually completely stop processing. Over a long period they would make a comeback even at that acid level but the time frame would be awful for a fishless cycle.
Another pitfall to avoid is the ammonia concentration going too high, up in the 8ppm range, and encouraging the wrong species of autotrophic bacteria. This can cause a setback that takes a long time to turn around.
Another problem for people sometimes is fishless cycling at the wrong temperature. We've honed in on 29C/84F as the temp of choice based on a number of bits of info and being way below or above that can be a problem.
Anyway, whatever the problem, its likely the members can help, or at least be there to discuss things with you!
~~waterdrop~~