Okay, to clear up some misunderstandings:
1. "Plugins" are files that are needed for most mods to work. You can view them as an addition to the base games' files. They contain the information that is necessary to implement new (modded) features into the game. Plugins have the file extension.esp, .esm, or .esl. They MUST be installed to the ".../Skyrim SE/data" folder. There are some kinds of mods (e.g. texture replacers) that don't need a plugin file, but most mods do.
2. "Plugins" must be "activated" in order to take effect. You must tell the game "load this plugin please". Usually plugins are activated through a mod manager. By default, this happens more or lesse automatically, whenever a new mod is installed.
3. Since you have installed mods manually, it's possible that you can't activate them with NMM. In this case, you can still activate the plugin (in your case KS hair) in game. Start Skyrim. On the loading screen, click "mods". The game will tell you something about a Bethesda.net acoount. Ignore this, hit "space" instead (default setting). Now you should see a list of all plugin files installed. Next to them there's a checkbox. If this checkbox is checked, the plugin is active and the mod should be working. Look for "KS Hairdo's" and see, if it has the checkmark next to it. If it hasn't already, check the box. That shoudl do the trick.
4. If you can't find "KS HAirdo's" in the "mods" overview, chances are you installed the mod to the wrong folder. All mods you download and install manually should be copied to ...Skyrim SE/data.
5. Regarding "download with Vortex": AFAIK, the Nexus doesn't force users to use a specific mod manager. This might be their way of advertising Vortex, but you should be able to download and install files with any mod manager of your choice. Unfortunately I don't have NMM installed any more, so I can't guide you trough the process step by step. I'd recommend clicking on the "download with Vortex" button anyway and see, what happens. Most likely you'll be asked if you wanted to manage downloads with NMM, to which you say "yes".
EDIT: I just checked some mod pages. On the very first page it says "Vortex" indeed. Still I think other mod managers should also work. But you shouldn't use that download option on the main page of mods anyway. A lot of mods offer different options and maybe additional files. To see all that stuff, you always should go to the "files" tab before downloading anything. This way you can see all the options you have regarding this mod. And on the "files" tab, it says "mod manager download"...