Measuring ring gaps:
First, you are only concerned with the area of the cylinder where the rings ride, not the ENTIRE length of the cylinder.
Put a ring in the bore, & using the piston as a tool, use the piston head to push the ring about 1" down in the bore (you do this to keep the ring flat in the bore). Measure the ring gap & write this number down.
Now, using the piston again, push the ring half way down the ring travel area. Measure the gap & write this number down.
Now, using the piston head again, push the ring to the bottom of the ring travel area. Measure the ring gap & write this number down.
If all 3 numbers are the same, you are fine. If the numbers change (bottom ring gap is larger than top ring gap), & you haven't measured an area of the cylinder wall where the rings DON'T ride, the cylinder is tapered & must be bored or replaced.
Since the rings stop at the same point on every piston stroke, the cylinder walls wear where the rings ride. At the top & bottom of the ring travel you will find a ridge. The top ridge is easy to remove (that's what a ridge reamer does). The bottom ridge is NOT easy to remove (that, & cylinder taper, are why cylinders get bored). If you didn't remove BOTH ridges, it's not surprising a ring broke. New rings are thicker where they contact the cylinder wall that used rings, so they WILL contact said ridge. If you're lucky, you only broke the ring. If you're unlucky, you also cracked the ring land (the groove in the piston the ring sits in) & that means a new piston.
The typos in this post are creating all kinds of problems!
BTW, rebuilding an engine without a factory service manual, is a crap-shoot at best! Unless you have certain specifications (like ring gaps, & torque specs & sequences) you are GUESSING & I can guarantee failure, sooner or later (& I'll bet sooner). If you don't have time to take a class at a vo-tec school, then find a couple engine rebuild books for beginners. The questions you are asking are really quite basic, & easily answered, but trying to learn this on the internet WILL end up being a lesson in frustration. Not to sound condescending here, but you need more than just "a little knowledge" to rebuild an engine correctly, because as they say, "A little knowledge is a very dangerous thing!" With that said, the BEST advice I can give you is to take the engine to someone that knows what they are doing. If you can find someone that's willing to teach you as they rebuild it, all the better, but I think you're setting yourself up for catastrophic engine failure unless you get a LOT MORE education on basic rebuild procedures.
The last time I rebuilt an engine was over 30 years ago, too, but I was taught, first in Vo-tech school, then by the US Coast Guard. With that knowledge, I'll happily send my engines out to be rebuilt because I don't have the special measuring tools, or clean space, necessary to do a proper job.