I have an NCO (inherited from another section) who just can't get right. Do you have any suggestions?

This is something that is pretty much common. Rather deal with the issue of having a bad NCO and reducing them to a rank that is more appropriate they will simply just move them to another unit and forget about the problem. First, I would give them a very indepth initial counseling that is referenced for the duty description and responsibilities with their MOS. Then I would give them the full load of responsibilities. If they fail they will be subject to counseling. You will have to build a packet. After such time you should have a pretty extensive folder. Another method you can use is to show their lack of competence to you battalion. When there is a duty for a NCO that is crucial you should volunteer this soldier for it. If your unit says no then ask why. If they say that he/she is not capable then you ask them for a reduction due to putting an undue burden on the other NCOs that have to make up for his/hers short comings. If they let this NCO perform the duty and they fail then you will be able to see the issue and they will have egg on their face also. It is not a pretty business but it is better to show them what they lack than have this NCO get someone hurt.

Not much to go on but it looks like you pulled your turn babysitting a rock. If your CoC is worth any salt, there should have been significant previous massaging. Your SEL should be very aware of this individual. So first, what's the plan? It had better be an up or out thing. If you're the last stop in the we're not letting him reup, then that's a different plan. If there wasn't much going on the past year or two, you'll have to decide if it's worth your energy pushing the CoC to do the right thing vs. getting dinged for showing up their faults. Get some insight upline lest you become the problem.