I love when people make lists for me of what Shakespeare stuff is available for streaming. IBTimes has 8 options to enjoy, with or without some company. Although in fairness, 5 of them are actually on Amazon Prime and only 3 on Netflix. Come on Netflix, step yo game up!
Shakespeare in Love is probably the most obvious choice (if you haven’t already seen it a dozen times). There’s that newish Romeo and Juliet with Hailee Steinfeld from a few years back. 
I tried that one once, but couldn’t get past the joust at the beginning. You remember that part in the original text, right?
If you like the older stuff, Amazon’s got the Helen Mirren / Judi Dench A Midsummer Night’s Dream from 1968, and a WWII Hamlet that I’d never even heard of.
There’s also some documentary stuff, including Shakespeare High that I remember for years ago but have never sat down to watch. Might have to check that out. Something for everybody!