Jacques
Yes a 7805 would give you a fixed 5v you could put into the 5v pin.
Get the data sheet and read it through. It will tell you in there that you need a capacitor on the I/P and another on the O/P. Use the values they specify.
The data sheet will show you how it connects, which would be, in your case, the +v from the wallwart to Pin 1, the -ve to Pin 2. Also a wire from Pin 3 to the Nano 5v I/P and another from pin 2 to Nano Gnd or 0v.
DON'T forget the caps between 1&2 and 2&3.
The 7805 will give max 1A (some claim 1.5A but treat all as 1A max). Currently you are going to draw approx. 100mA. Let's say the wall wart, at this point, is actually giving 9v. You will therefore be dissipation 4v*100mA = 400mW. Nothing to worry about. BUT what are you going to add to it and what current will they require. That will increase the dissipation and therefore the 7805 could start to get hot and need heatsinking
Better to buy a cheap buck regulator from ebay or other suppliers. Arduino Shop???
Connect the wall wart wires to the buck, attach DMM to O/P. Adjust onboard trimpot to give 5v O/P. Connect to Nano. No heat worries.
Good luck
Fof