Replacing rear hanger — will I need an alignment tool?

I have replaced countless hangers on various of my bikes road and MTB as well as my son's bikes for donkeys years. I have never needed nor do I own an alignment tool.

As others have said, if the hanger is bent or broken then it has done it's job of protecting the frame.

Unscrew the deraileur from the hanger
Take ithe hanger off
throw the bent hanger away
put a new hanger on
bolt the derailleur to the new hanger.
ride your bike and smile.

Depends on how heavy the 'off' was that wrecked the previous hanger? I have experienced frame drop-outs being distorted by the impact on a rear mech and needed to be 'adjusted' such that the new mech hanger was straight. Suggest you fit the new one and if the rear mech alignment is visually OK then it should be fine. If however you have a problem with gear adjustment, particularly the biggest sprocket, then it may need a tweak.

As has been stated the hanger is a disposable component designed to bend/break before the drop out plate gets damaged.

Your Defy has a fairly chunky lump of cast then machined aluminium forming the drop out plate so that should not be out of line unless you managed to bork the rear triangle beyond repair.

Most replacement hangers are cut from T6 ally plate, fitting a new one you don't really want to go bending it again to make it work, if you have to bend it then you may wish to consider a new ally frame.

A derailleur alignment gauge is always nice to have, but in this case you should be OK.

If the frame was not bent during the incident during which the old hanger was damaged, then you should be OK. Just pop in the new hanger and all should be fine. This is the most likely scenario.

You can also align the hanger by eye. Firstly make sure the hanger is tightly attached. Then hold the derailleur body (not the cage) and use it to bend the hanger until you can see the cage and rear cog are aligned. Obviously this is not ideal.

Another method is to remove the derailleur, and insert your wheel by screwing the axle end into the hanger. You can now use the wheel as a lever. You then measure the rim distances until they are the same (the wheel is aligned). This will automatically ensure the derailleur (when reattached) is also aligned.

Or just take it to a shop...

you shouldn't need one as it's the hangers' job to protect the frame ....