The Dudh Koshi is the classic run from Everest. Basically can you start at Namche Bazaar (confluence with Bothe Koshi from Cho oyu), where you can Kayak all the way to Gopeghat mainly IV+ (V), then there is a major portage around the gorge at Basa. From there onwards will be more grade IV+ (VI) with the river easing down until it reaches the Sun Koshi.
A great river with stunning views from the Kumbhu Himal. The logistics aren't that difficult, as the trail follows the river and Lodges are plenty to be found. If you get the boats carried in from Kathmandu, the costs won't be to bad either.
Look at a long trip, which you can combine with a trek to Everest Base Camp and will get you all the way down to the lowlands of Nepal.