After removing the broken zip-tie and replacing it with a sturdier one, I adjusted the float level and thought I was good. I wanted to get the water level a bit lower, but the float type wouldn't allow it, which bugged me, but I was happy to have the leak fixed. I left the lid off so I could monitor everything for a while. A day later, I discovered the water level was very slowly rising and running into the overflow pipe. Who knows how much water was wasted.
I happened to have a spare valve on hand because I unnecessarily replaced it in the other bathroom. Long story, that I won't go into. Oddly, I can't find a blog post about it.
The first step was draining the tank and removing the old valve. I recommend having plenty of old towels on-hand.
Once that was done, installing the replacement valve was a cinch. So far there's no leaks and I was able to dial in the exact water level I wanted. Success.