Dog Chewed Through Power Couch Cord

It should have 3 wires. Pretty easy.

Should be able to splice it fairly easily, or does the chair just use one of these

I would just splice the wire.

To do it "correctly" I would use Western Union or even better NASA splices (YT had videos) solder and heat shrink the splices. I like to do each conductor offset so there is no chance of shorts and then put larger heat-shrink over the whole area. The WU or NASA splices are very good at preventing the wires from being pulled apart. Maybe overkill in a cable that sits in one place but it makes for a permanent solution where something like wirenuts are easy to fail in this type of use. I have don't it on outdoor extension cables cut by string trimmers and saw and they have lasted for years being pulled around like you often do.

It might seem a little much to learn a new skill and buy a soldering iron but the kits are cheap and being able to solder and do proper splicing is a really useful skill that comes in handy a lot once you have it in your pocket.