The Command Prompt still works fine.
So you should try starting Powershell from it. E.g. in an elevated cmd window you would expect to see it converted into an elevated Powershell. What happens? E.g. does the whole thing then go away?
Another approach along the same tack but with potentially more control and better diagnostics would be to use ProcExp to start it. E.g. ProcExp's has two choices of Run as...
In fact, ProcExp's Search tool (Ctrl-f) for System.Man is showing me that other processes besides Powershell would want to load various instances of System.Management.ni.dll from %windir%\assembly\NativeImages_v4.0.30319_64. One of them is mmc.exe which I guess must be Hyper-V Manager. Vmconnect.exe is another which would be related to that.
Oh. In fact, it is probably System.Management.Automation.ni.dll that you are missing. Only Powershell and its ISE are loading that.
FWIW I always start ISE from PowerShell but I'm not sure that I need to. So you could try starting it independently instead.
Otherwise, I wonder if those .dll are checked by SFC /scannow? That might repair them or at least give you better clues about what is wrong with them. Ditto for DISM as a potential repair/diagnostic.
Good luck
Robert