Me personally, I'd unplug from everything and only allow myself like 30 minutes each day to check for any critical emails (like oh shit type emails, ignore the normal ones). Then I'd spend my time talking to real people and find things where people are frustrated where I might be able to make an impact. Basically, spend the time learning from them and spending time understanding problems they have (not Dr Phil style of course, but everyday crap). People hacking (for learning) in a sense.
But assuming you don't need/like that then here are another few ideas:
1. Learn the basics of how to architect a small distributed system that is fault tolerant, resilient and reliable. You can't learn it all in 11 days, but man you can make a big dent and it is to me one of the most fun and challenging problems to solve. And there is still so much to learn. You can build and test little systems in AWS for nearly nothing using t2 instances.
2. Do as iqonik suggested, find a simple need some group of people have and solve it with an MVP and start marketing and playing with it. If it goes nowhere but you solved an interesting problem for even a small group of people you will still feel good and have had some fun.
3. Write an Android app. From the looks of it you haven't done that but it seems basically up your overall skill set and something you could have fun with. Even if you "hate" Android learning how it works and the good/bad sides might help you in other places.