SGS2 screen gets more and more dark blue, possibly after overheated charging
Looks like a damaged LCD that's slowly "leaking out."  Overheating could definitely be the culprit, as could the drop. Although damage from the drop is usually accompanied with a visible crack.  Most likely it was the combination of the two.  The fact that the phone booted up and is (sort-of) functional leads me to believe that replacing the LCD assembly would fix it.

There are plenty of replacements to be had on eBay for up to USD$100.  I would go for the LDC+digitizer combination, even though the glass itself looks intact. There are simply too many issues with properly separating LCD from the glass that could cause you to break both and spend even more money.  It is definitely doable on your own, provided you have some basic tools.  There are videos on YouTube with great walkthoughs to help you along the way: probably most detailed (even though it's not on GT-I9100, but on other modification of SGS2); some others specifically on GT-I9100. A guide at iFixIt can be also helpful.