I've drilled through a LX cable running to a light switch... what to do next?

It is illegal to splice cable unless the connection is accessible. You may end up with two outlet boxes if there isn't enough slack to do it in one box, maybe a 4" box with a blank cover if you are lucky. Some people will want to replace the cable, that means doing a lot of damage unless they can re route the cable to the outside or something. This is where innovation and experience can complement one another. Good Luck.