You'll have to replace the whole handle now and you might as well do the otherside while you're at it. You'll have to remove the interior door panel also to get to the door handle so you may want to check up on some other things while you have that panel off, like are your windows closing properly, how's the door lock knob, pwr mirrors, etc etc.
Get the steel replacement handles while you're at it. Forget those plastic one. They come in black and crome. I think the crome ones are aftermarket and 50resto sells them.
As for using rivets, you can buy the rivet tool/gun for $20 at an auto store. they usually come in a kit with 4 sets of different sized rivets so you should be able to find the size you need. In fact, I did some rivet work on my car and didn't want to keep the rivet gun, so I just took it out of the package real carefully, used it, and then re-packaged the gun and returned it for a full refund!
Using a nut/screw and bolt with a washer is an ok idea, but I'd be worried that over time that nut will get loose and the only way to tighten it is to get behind the door panel, and taking it off all the time can get tedious and tiring. Rivets tend to hold well and if "popped" in correctly, will not work their way loose over time. If you do use the nut and bolt idea, ask for something called a "lock-nut". These things are used on stuff like skateboard truks and wheels. They allow the wheels to spin without loosening themselves over time. They have some rubber or plastic gasket inside them.
Then i went to Ebay. 
