Do I have any misconceptions about what an internship entails?
I was expecting to have a well defined project ready for me to work on and a mentor that will proactively offer guidance and check in on me since they opened up this position for one reason or another.
I honestly assumed you were being funny when I first read that!
OK, so now you have completely, totally, utterly learned that you were wrong to think that, and it's not how the world works.
I've thought a lot about the best advice to offer here. I believe it is this:
You have learned that two weeks is an incredibly long time.
Every single person reading this, will be dead in 5000 weeks.
You can not waste two weeks of life like this. After two hours you should have acted, and urgently.
You must act (now, i.e. after reading this).
Walk over to the manager and say "Thanks boss, you're awesome. I finished all the odds and ends. Can you give me a really difficult programming task and I will get right on it, and give you blood sweat and tears on it - let's go! What's the toughest most annoying thing you need done, I'll do it."
If you don't get anything (and that's their prerogative - it's a free world, nobody is owed anything), leave.
You can achieve an incredible amount in 10 weeks; an instant obvious suggest build an open source widget that will give you excellent publicity the remaining 1248 weeks of your career.
(Your career in this lifetime will be 1250 weeks at best; you've very unfortunately wasted two, or had two wasted for you. But you've learned a super-valuable lesson: never waste two weeks. Stick to the "two hour" rule. If anything is proving to be a time waste after two hours, walk.)
Note - when you leave (assuming you don't get the Tough Project, which is possible), be super polite. Here's some great language: "Boss, you've been awesome. I can tell there's nothing for me to do here - the company was too kind in offering an internship. I've stumbled on to a great open source project that will really stretch me and fits in to the next 10 weeks. Thanks again, it's been fantastic, I'll clear out my pencil from the desk."