Never have spliced wire before - can I splice the wire without soldering?

Sure you can. It's not hard at all.
Cut your existing wire, then split the two halves back an inch and a half or so. Using wire strippers (or a sharp knife / razor blade) remove the insulation 1/2 from each end. Do the same thing with one end of your "extension wire" Twist the wires tightly together one conductor at at time, then wrap with electrical tape. Make sure you tape each "half" of the wire individually to insulate the two conductors, then wrap the entire splice w/ electrical tape to protect the splice.
Repeat the above for the other end.
You can make your splice a lot neater by using shrink tubing instead of electrical tape - just make sure to put the tubing over your wires BEFORE you twist them together. Then slide the tubing over the splice point and heat gently w/ a lighter until it shrinks tight.
Dan