The DM-22 is chambered in .22 Magnum (.22 WMR), so of course the firing pin should be striking the rim. If it's leaving visible strike marks on the rim, yet not firing, the first thing to do is to buy a new box of a different brand of ammo.
If it still doesn't work, then you have the classic symptom of a primer strike that is too light. That could be from a weak spring, or it could be because gunk is impeding the hammer from properly striking the primer. If it's "old" as you said, chances are it's full of pocket lint and dried oil. Spray the hell out of it with solvent, lightly re-oil, and try again.
Just in case your brother doesn't know this, .22 LR and .22 Magnum are not interchangeable. The Magnum has a larger diameter case, so .22 LR can split when fired in a Magnum chamber. It's not a problem in a derringer, since there's no real back pressure due to the short barrel, but extraction is a problem.
By the way... is he trying to shoot it with .22 LR? The difference in rim diameter (WMR = 0.294, LR = 0.278) could result in a misfire.
Caliber markings mean things.
