Take out your DMM and make sure what color wire is hot and neutral through the power cable. Then remove the cable and get a 3 prong type power cable. (Assuming you are in the US) Basically, you have answered the solution all ready in your post. Drill a hole in the chassis and attach a ground wire. It is really easy as you said in your post. Sometimes I find Black hot and other times white is hot...? Look at the function of the power cord in relation to your DMM. Adding a ground wire should solve your problem.
P.S. - Star grounding is something totally more advanced in my thinking in relation to just adding a simple ground wire.
P.S. - Star grounding is something totally more advanced in my thinking in relation to just adding a simple ground wire.