Ok, here's the problem Jovial: leafminers are larvae that are eating the leaves from the inside out. The only thing the neem is doing for us is, hopefully, killing eggs laid externally, as well as any pupated larvae that have come out of the leaves. There are no good systemics that I have been able to find that will kill the bugs once they're inside the plant, and that's what I was hoping for, something that would be taken up by the roots and go into the plant tissues that would stop the attack dead.
Also, I'm sure you have learned that there are MANY leafmining bugs, not just one. I am pretty sure I've now found the culprit, but not positive. I could talk to our county agricultural department, but there's no way in hell I'm bringing them a leaf sample. I have found two tiny grubs, each on one plant, but I killed them both. I can "hope" to find others, and probably will, but by then the damage is done. However, if I can find another grub I can take that to the ag department and possibly get at least a positive ID, then try to encourage those insects that are parasites to the bugs that lay eggs that lead to leafminers. I hope I'm making sense.
I've got some links if you like, but so far nothing is definitive in terms of helping gardeners like us that I've found so far.