The problem that I am getting happens without pressing any key. Simply ssh in, run vi/vim /directory/file to a known file and the vi/vim session goes into a condition that can't be controlled with any keyboard input. This only happens with KDE4 Konsole and also only happens when connecting to a Linux system that does not have the x window system or any xorg apps installed. But, xterm works perfectly. The condition also will not happen from KDE 3.5.10 on PCLOS.
This is similar to what I tried; I used vim to open /etc/fstab over ssh. If I tried to edit it using the insert keys it would do odd things, but if i used the letter i to switch modes I could then add text normally using the keyboard. some of the "extended" keys would still behave oddly (some of the arrow keys -- which I believe are only enabled in newer version of vim -- would not move the cursor, but instead create odd entry).
If I just navigate to the location I want to type using either letter keys or keyboard, hit "i" to insert/edit text, then hit "esc" when I need to move the cursor or enter vi commands (ie ":q!") to quite without saving changes, it seemed to work as it should.. from what I remember of vi/vim, this is more of the classic way of using it-- I think it was initially designed so that most everything could be controlled from the "home row" of the keyboard.
It sounds like you are working from a graphical environment... perhaps a better solution would be just to use ssh -y username@server-ip
and then open those remote documents in kwrite / kate, etc.