Wow, I'm surprised it was so complicated. It really doesn't seem to be much beak at all. I have a hard time imagining it being so much trouble. I know Capes have long and sensitive beaks, but that was such a tiny piece that was busted. From your description it sounded like a chunk the size of maybe halfway from the lower beak to the tip was broken. So when I saw the picture, aside from cutting the remainder off it didn't seem so bad. I still think I would have waited till at least tomorrow to see if this takes care of itself since there's no bleeding or further damage.
Here's a video of how I was hand feeding Truman and how I gave him medications. Basically you just squirt it in the beak and the rest is up to them. You can't make them swallow it and they are likely to shake it all around. But some may still stick to the tongue and make its way in. Try taking the water away for some span of time and serving a sip of water right after the meds to help flush them in.