The blog has refused to save anything above the next line twice in a row - and I tried manually saving it as well.
That really makes me upset.
This word is something that you'll encounter at least once if you work in Japan for a period of 12 months or longer. It literally translates as Health Check, and it's something every working man or woman in Japan has to go through. My company finds the lowest bidder health service, brings them in, lets them listen to our heartbeat and has us cough a couple times, then sends us on our way. They have about as much chance of finding something wrong with us as dogs have of seeing the world in color.
The short version: 健 means "health," 康 means "health," so you have "health health" or some word that very likely has something to do with health. Imagine that.
診 means "investigate, analyze" and 断 means "end, finish, terminate," so you can assume that they are trying to infer the doctor analyzes your fully and completely.
And this pair might seem a bit more difficult than "health health," but the first kanji holds power over the whole word (true in many cases - that's why it comes first, to set the tone as it were). So "analyze or investigate" from the second line coupled with "end, stop, finish" can be interpreted as "checking thorougly" or "leaving no stone left unturned." So without being able to read these kanji (and I admit the first time I saw them I had absolutely no idea beyond "ken" how to pronounce them) you can understand what it means in context.
Especially when there are a ton of nurses and doctors standing around and people waiting in line with small cups...