Slipping in an external adapter
12V 0.9A means 12V 900mA, not 9mA, which is 0.009A. When replacing a supply, always go with the same voltage and equal or high current capability. So 0.41A is less than 0.9A, meaning it can't be used.

And yes, you essentially have the right idea. Cut the old +12V DC power supply wire out, splice the new +12V DC power supply wire in, and tie the DC grounds together.