Try doing this in order - The order matters !
Go to Nighbot control panel. Part it from your channel and then LOGOUT.
Go to Twitch and LOGOUT.
Log back into Twitch.
Log back into Nightbot.
Join NB to your channel.
If this doesn’t work, try deleting cached content and cookies for Twitch and Nightbot after logging out from both again. (You may have to reset pages back to how you want them again e.g. chat settings on Twitch).
Let us know how you get on 