1.You need to go to the options button in the bottom left corner, then the 'Inventory' button in the 'options' menu. The one where you select the modes of 'Cheat', 'Recipe', 'Utility'.
2.Besides doing this, you need to click on the 'global' button, after you do that.
3.'Right click' the 'W' next to the mode button. This restricts the magnet utility/cheat functions being displayed in your inventory menu. Right click the W and you should have the cheat functions or magnet only (depending on if your using cheat or utility mode of course, not recipe).
Hope this helps. I remember it every time I use the later 1.7.10 and 1.8+ NEI versions, it I do it a lot and its stuck in my head since.
4.Now to enchant an item, you need to press the 'X' key by default to enable the enchantment menu. The potion menu is 'P' I believe.
NEI requires you to have JEI loaded for some reason and you can toggle the interface back and forth between the two. I installed NEI (which I thought was defunct a while back, but apparently not so much...).
I couldn't get the enchanting menu on JEI either and I used to run it a while back and I wasn't even sure I could remember that there had been one. I lost a game where I had a ton of awesome enchanted items that I got legitimately and I wanted to restore them without having to grind out millions of experience points, so I thought I'd cheat them in to my new game (which was actually a restore of an older save file from before I had those items).
JEI doesn't have any of the NEI features that I know of, only the recipe/cheat mode as well as extra search methods like @mod for example. Covers, the new author has taken a lot of time on the NEI, Ender Chests, Translocators and other mods compatibility very well. Just no Wireless Redstone ChickenBones Edition obviously. At least from what I can tell from the 1.10.2 to 1.12.2 versions since 1.9.4 was a bit messy. They work a lot better than the other alternatives I've come across like Forever Enough Items, Ender Thing, Ender Tanks.
But I think NEI is build on top of/along side JEI since its a lot of effort to make NEI stable, I know at least in 1.7.10 it was difficult at times for me when using or just getting the mod to work, and since JEI is the way to go for recipe mods as well as its got the popularity behind it. Its best to do so than a whole new one and the addons are already for JEI. But that's just my theory of course.
I wouldn't call it cheating if your working around experience and enchantments to restore what was lost, if the backups don't work out. I've used the enchantment option/function as well as the potion one mostly in 1.7.10 and never again since I don't use it for adventure map making anymore. That's the only time I ever used them from memory. But they work well and even I think allow objects as well to have enchants on them and properly add Fifth or V enchants nicely.
Your welcome. Just remember that CodeChickenLib and CodeChickenCore are switched around this time. CodeChickenLib is the one you need as a higher priority. This is more obvious with mods like Thermal Expansion using it of course, but if you weren't sure of the change, now you know.