If the damage (burning/melting) extends inside the multilayer circuit board, then it will be very tricky to repair.

You could try the simple thing of replacing the two obviously blown components, and then powering it up using the indoor AC-adapter. Most likely, they will blow again, but you could get lucky, and have almost nothing to lose.

If that doesn't fix it, then I recommend putting the unit aside for use as "spare parts", and trying to purchase a replacement unit. This should be cheaper than paying for a difficult repair job, and the unit may be "beyond repair" already regardless.

But first I'd try the suggestion of replacing the two blown parts, just for fun.

-ml