Give a try to Dependency Walker to learn if you are missing required files.
http://www.dependencywalker.com/
I have not used this for a few years, but I remember using it a couple of times to hunt down missing dll files.
Good luck!
PS---Also, have you tried having Task Manager open while Sigil launches to watch what happens? I have one program that behaved similarly to what you report about Sigil, and it didn't actually crash, the splash screen disappeared and the app continued to try and fail to load in the background. My solution for that one was to always Run as Administrator, because the program was being prevented from loading a dll into memory and got stuck at that point. The admin rights let it load what it needed.