if the power adapter was plugged in while the dog did this, there is a chance that the power supply burned itself out from a possible short and needs to be replace. if it was unplugged or is not internally damaged (besides the chewed cable), then it is theoretically possible to repair the cable.
you can splice it or solder the cable back together, but your handiness with things electrical will dictate how reliable the repair is. the power going through this part is low voltage ... DC. if for example you do a shoddy repair and don't have proper strain relief, the repair could short in the future and potentially cause a fire. i would do it, because i'm confident in my abilities, but i would still buy a new adapter because i'd rather not travel around with a repaired power supply... and keep the old one as a spare.