There's an old saying that "experience gained is in direct proporion to equipment ruined" (I'm living proof of that). Since you have access to a parts diagram on-line, look at what the "piece at the end" is actually called (probably a "check valve"). Then go and order a spare set (it's a good idea to have spares around in any case). When they come in, install them (carefully!) and pay attention to any differences with what's in there at the moment.
Yes, many check valves *do* have a plastic sealing ring or spacer, and if it's missing, the valve *will* leak. It's quite possible that someone before you cleaned the check valves and didn't put them back together correctly.