I always love how people's playthroughs of these types of games differ.
You defeated the Archbishop in a few goes and he took me about 30 tries and grinding some levels, and yet I killed King of Puppets on my first go
Best thing I can suggest is a full load of throwables, but save them until you have him down to around 50/60 percent on his second stage (i used this method on quite a few of the latter bosses).
Also, get the cube buff that heals the Spectre as having them for the second phase is (literally) a life saver.
Don't give up skeleton!