It's impossible to use registers in VI mode in Kile. If you press the " key you enter INSERT mode and 2 backticks are inserted. It's like the vi command is not ovverriding the Kile behaviour. In Settings -> Editing -> VI Input Mode the box "Let VI commands override Kate shortcuts is checked". In Kate registers work fine.
I found a duplicate: https://bugs.kde.org/show_bug.cgi?id=247032 Sorry for this report, but the other guy was using "buffers" instead of "registers" and I did not managed to find it. *** This bug has been marked as a duplicate of bug 247032 ***