Langtang is a great 7 day trek, Kathmandu to Kathmandu. It's a long bus ride though, each way (10 to 14 hours). However, you get to experience the culture on a bus and you get to see lots of scenery. You get to see some decent size mountains (7000 m) on the trek, but not quite as high as the 8000 m ones in the Everest or Annapurna regions. You trek through several villages where you can get a great cultural experience.
You could also do a part Everest Base Camp trek. In 7 days, you could get up to Tengboche or Pheriche and back, depending on how long of trekking days you want on the return. However, remember that you could be delayed flying into or out of Lukla. Flights operate on visual flight rules so if the weather is poor, the planes don't fly. People have been stranded in Lukla for up to 7 days. However, this is on the long side. So, if you are going to do this trek, do it at the beginning of your trip so you have extra days in case you become stranded. You don't want to miss your international connection. If you don't have any delay, then you have lots of time to see the sites (including the world heritage sights) in and around Kathmandu and the valley.
You could also take a bus or fly to Pokhara and do a variation of the Poon Hill trek over 5 days. Poon Hill, if its clear, will give you a wonderful panorama of mountains in the Annapurna region. It will give you a good cultural experience too because you'll be trekking through numerous small villages. However, this trek will be crowded with western trekkers trekking down off the Annapurna Circuit. If you go to Pokhara, spend a morning going up to the lookout at Sarangkot if its clear. Your hotel can arrange for a micro-bus or taxi to take you there for a sunrise visit. It gives you a wonderful view of the entire Annapurna range and you can even see Dhaulagiri (another 8000 m peak). This lookout area will be very crowded if its clear, but its worth it.
I've done all these treks. I would recommend the Langtang trek for your time frame unless you just have to see Everest. On the partial Everest Base Camp trek, you'll get a distant view of Everest, but not real close. However, there are other in-your-face mountains like Ama Dablam. The Langtang trek won't be as crowded with trekkers as Everest.
