How to waterproof siding repair?
You have to chose between cosmetics and functionality. Either way you're still going to need some more siding. 

Functionally, you need like 2 pieces worth and you could remove and then stagger the existing. Cosmetically, you'll need most of a box. 

There's supposed to be a metal shim and caulking in between butt joints. Without entirely removing or prying away (can of worms) the siding at the corner, there's no way to waterproof it.

Super awesome exterior caulk is made by Vulkem. Which obviously works pretty well because it's known to the state of California to cause cancer, birth defects or other reproductive harm.