VGA Monitor cable cut with a knife... monitor still salvageable?

Do I understand it correctly that the VGA cable is fully attached to the monitor, with no way to deattach it?

You could probably get another VGA cable, remove one connector at the end and then try to attach it to your monitor. That would of course involve opening the monitor casing, desoldering the old/broken cable and soldering the new cable to the now open contacts. I don't think that opening the casing and desoldering is the big problem, but since the wires in the VGA cable are rather tiny it could be a problem when you want to solder the new cable.

If you have soldered microelectronics before this should be definitely doable, otherwise get yourself someone with the needed experience.

Or maybe the monitor has additional connectors, like BNC?

Jam it back in, in the dark.

However there are some problems with the methods:
(i) Just attaching a new connector to the cut cable will leave you with a very short cable. Now you need an extension cable. Extension cables for VGA -> bad. Quality of transmission on standard VGA isn't good at all. Introducing even more plug/jack "changes" isn't going to make it better, it's best to avoid that.
(ii) Same goes for soldering another cut cables to the current cut cable. Introduces another point where cables/contacts are connected via soldering, which will increase resistance of the whole connection.

Take your pick.

There's nowhere I can't reach.