Standard laptop power supplies are 15V 5A. This is sufficient overhead (even with a voltage drop from current draw) for an LM7810 (or any other 10 volt linear regulator). I suggest running a 5V regulator off the 10 (7805 or 78M05) instead of relying on the small internal arduino one. If you do choose to supply power to the arduino, use the regulated 10 volts and not the full 15, that will overheat the regulator with any reasonable current draw.