Should be fine, the old one can provide more current than the new one so won't be overloaded. If the ratings were the other way round it could be a bit risky.
Main thing is to make sure the polarity of the output jack is the same, it generally is for a given manufacturer.
As for the bare wire- is it on the mains or the DC side? The reply only affects the thickness of insulating tape you need
Alternatively, cut the damaged section out and use a suitable in-line cable connector to join the ends up again.
Something like this
would be fine- it's a bit overkill if it's on the DC side but that doesn't matter.
Edited by billford (Sun 21-Dec-14 13:06:14)