How do I fix a broken power cord
Since the damage is at the end of the cord, you can just cut off the chewed part, replace the plug, and lose a few inches of cord length.  There is an endless selection of replacement plug styles at any big hardware store.  A few things to note:


This answer is focused on the plug style used in the US, Canada, Mexico, Japan, and some other countries.  Plug designs vary in other countries.  To the extent that there are other considerations that apply elsewhere, I encourage readers to comment or post other answers to better address requirements in other countries.
The cord in the picture is commonly called zip cord or lamp cord.  It's used on low power, indoor appliances.  Plugs are designed for different types of cord.  Some cords have a round outer insulation jacket covering individually insulated wires.  The replacement plugs designed for that type of cord, especially for higher-power appliances or outdoor use, are big and contain a clamp sized to grab a thick, round cable.  You don't need a massive plug like that.  There are light duty plugs designed for the type of cord on your air purifier.
It's a two-wire cord, so get a two prong plug.  There are polarized and non-polarized plugs (the polarized plugs have one blade wider than the other).  Match the style of the original plug.
If the original plug was polarized, the cord should have one side identified by ribbing or a white stripe.  That side goes to the wider blade, which is the neutral connection.  If the original plug was not polarized, it doesn't make any difference which wire goes to which non-polarized blade.  
If the store has only polarized plugs and the original was not polarized, it won't hurt to use a polarized plug.  But if the original was polarized, do not replace it with a non-polarized one.
The plugs come in two attachment styles.  One requires you to strip some of the insulation and wrap the wire around a screw.  The other is "insulation displacement"; you put the intact wire in a slot and closing the cover forces the wire onto contacts that cut through the insulation and grab each wire.  Either style is fine.  As FreeMan notes in a comment, some insulation displacement plugs may require you to separate the two wires for a short distance, with each wire going into a separate channel inside the plug (just follow the package instructions).  






Make sure there is a mechanism to secure the wire so that pulling on it can't pull it out.  If there is not some type of built-in clamp (a situation more common in the screw-type, like the plug shown below), tie a knot in the cord inside the plug to keep the cord from being pulled out.





There is a knot called the Underwriter's Knot designed for this purpose.  The picture below (courtesy of this web site), shows it inside a lamp base:





You can get right-angle replacement plugs.  This can reduce the clearance required at the wall if the outlet is behind furniture.




all pictures courtesy Home Depot