Recently, my new puppy in the house chewed my laptop power cable in half. Can I splice it back together and if so, is it safe?
Absolutely, if you can uniquely identify each of the individual wires in the cord. If you can't be certain what to connect to what, don't guess.
If the cut is on the cord between the brick and the wall, every hardware store sells kits to fix extension cords, which will do exactly what you need.
If the cut is on the cord between the brick and the computer, it is a bit harder. I suspect if you knew how to solder and shrink wrap you wouldn't have needed to ask here! Find one of those TV/radio repair shops that has been around for 40 years. The old guy that runs it will know how to fix the cord, and you will probably have a good time listening to his stories while he does it.
You can if you want - chop out the chew then connect/tape together... I did this years ago on a damaged cable.
However, personally, now that you can get them so cheap, I would just go to eBay and order another one! It isn't worth the risk and it just isn't worth the risk of something bad happening.