It sounds like you’re not using flux, which would cause exactly the
symptoms you’re describing. If you flux the join between the two
butt ends of the jump ring, the solder should flow between them. The
ring ends must be touching (solder will not jump a gap), so if they
aren’t you will also want to tighten them further using your pliers.
Don’t worry about getting them “out of round” at this point – you
can always pop them back on a mandrel of some type to re-round them
later. Even really tiny ones can be re-rounded on a pin, if
You can buy commercial flux through Rio, Hagstoz, or any of the
jewelry suppliers. Dandix, Handy, and Grifflux are 3 of the better
known brands, but there are many out there that are suitable.