soldering iron and solder + electrical tape should do the work.
have you ever used a soldering iron? It's not difficult if you're not worried about looks, but you have to kinda learn how to use it.
Maybe youtube has a soldering iron How-To?
Hand twiste the wires really well together, I'd say at least 1/2" of wire.
Melt the solder evenly onto the bare wires by touching the tip of the soldering iron to the solder, and you can use the drip method onto the wire.
Once you're sure it has a solid connection, tape it all up =) I think that's all it takes.
Goodluck!