1. Buy solder and flux and read the directions. "Michael Percy's" website has a good selection and reliable information. The 3 or 4 types that come up are all reputable and as usual folks have their preferences and reasons. If I were you I'd buy the solder with a low melting point as ease of use is what you obviously need; however any of them will provide a satisfactoty result-if used properly. Percy is reliable.
2. Prep the joint spot on the tweeter.First remove the old wire where it it attached to the tweeters by bringing the hot tip(preheat the tip) of your pencil/gun(hereafter called an 'iron') to the joint. Touching the joint heat it quickly until a gentle rub from a piece of cloth causes the old wire to be released. I like to remove the old solder at the same time rubbing the cloth on the hot spot. The old solder does not have to be fully removed or perfectly clean. (Your pencil/gun "must" be hot enough so this is accomplished in a few second or you risk heating areas nearby-which you'do not' want to do- this is what others are warning about. "If" your pencil will not heat the target quickly(i.e. to not heat up surrounding area) then I would obtain(rent) a gun with a hotter tip.) Quickly reheat the spot until a dab of the new solder wire melts onto it. This piece is now ready.
3. Now prep the wire end. Cut the insulation off the tip of your new wire - perhaps an inch. Preheat you iron. Heat the end of the wire quickly until touching the solder to it results in a drop of solder melting into the tip of the wire. Remove heat as soon as the drop melts into the wire.The wire is now ready.
4. Align and touch together the 2 items to be soldered exactly the way you want them to be permanent.
5. Bring the hot iron to heat the 2 items simultaneously until the solder wire touched to them melts a drop which fuses the 2 together. Immediately remove the iron and allow joint to cool naturally - without disturbing it for the 'say' 5 seconds it take to stabilize. You should now have a stable soldered joint. I hope this makes sense for you. Feel free to ask questions so we get you back to enjoying your system. Best wishes. Pete