Why didn't you just grab the professor suddenly, and use the invisibility potion of both of you...
:/
The effects of a potion are at their maximum when consumed at a certain amount, any less could cause nothing.
Put your cloak back on.
GO INTO HIS BEDROOM
The door to the house is locked! Looks like you'll need to find your way around town, explore more or something. You could also get outside Dalore while the invisibility is still under effect, sneaking past the guards.