I'm about 4 hours by bullet train south
Kaifeng, the ancient capital of the Song dynasty. It's relatively small by Chinese standards and you can see all of it in a day. This is a good time of year to visit Beijing, as in the summer all the Chinese tourists flock in and during the winter it's too cold, so October is a sweet spot. Forbidden City and Great Wall are the obvious must-sees. I'm sure your friends can show you around. You've got 16 days, the bullet train system in China is the most extensive and maybe the best in the world, so you can easily do quite a bit of traveling around the north. I've traveled quite a bit around China and can give you some recs if you plan on seeing a bit beyond Beijing.