You can solder them after using a western splice connection. You can use a mechanical butt splice and you can connect the wires together with a wire nut after twisting the wires together. All you have to remember is that the insulation of the splice has to be equal to the wire insulation or greater.

There is nothing wrong taping a broken wire. As long as the wire connection is made with an approved mechanical device, the splice can be taped. With broken wires, usually the conductor is put together again with a butt splice. On larger size conductors the butt splice is just bare metal. This type of splice has to be tapped. The code rule states that the tape insulation must be brought up to the insulation value of the existing insulation factor.

One splice that would work is a western splice.

The four types of splices and joints are the Western Union splice, tap splice or tap joint, fixture splices or fixture joints, and rattail joint. The most common among the four wire splices is the Western Union splice.