I'm trying to remove the left crank arm. I've previously been able to do this with a 10 mm Allen key. But as you can see, this time it's stuck. Does anybody know which tool is best for the job? I fear that I've ruined the bolt, so now there's no grip. Any suggestions would be appreciated!
I think your crank arm will need replacing - your best result here is getting the old one off without damaging the rest of the bike.
Start by spraying some penetrating oil in to help relax the thread's grip. Do it the day or more before to allow soaking in time.
I'd normally start with a large Torx driver, one that is oversize and will bite into the corners as its hammered in. However 10mm is large for torx, this is more suitable to 3/4/5mm hex bolts.
Some people have had good results by laying a wide rubber band over the fastener then fitting the normal driver. This is more for screws but might work here.
More likely to work is using an 11mm or 12mm hex driver, but shape it to a slight taper. Again it's going to use-up the tool, but you may be able to hammer it into the bolt with enough grip.
These are all one-time solutions - its coming off and never going back on.
Anything harsher would involved grinders or hacksaw - cutting the crank arm off, or through the BB axle and replace the bottom bracket cartridge as well as the left crank arm.
You might need to look at your 10mm tool and see if its undersize, or if you were routinely overtorquing this fastener. It should not have failed like that.
It's an FSA crank, not Shimano.
Presuming this happened in the first place because it's in there tight or corroded, you'll likely never be able to get the crank bolt out with what's left of its original wrench flats. It's hypothetically possible you could use thermal shock, chemicals, and/or impact to get the breaking force needed down and then either create some kind of new tool fitting or use an easy out and then unthread it like a bolt, but probably those sorts of plans will fail.
The plan here should probably be to cut the bolt out. Look at the FSA part number on the other side of the crank arm to get the model number and get the correct replacement. Once you have that you can study it to get a sense of how to plan your cutting so you don't damage the crank. You're choosing between whether to cut the head off the bolt off to relieve the thread preload or whether to try to section down the center of the (presumably hollow type) crank bolt. Probably you should go for the head and hope that once the bolt shoulder is no longer in place, you can get the shaft section out easily with an easy-out or improvised tool. Sometimes this as simple as just selecting the right size drill and aiming it right in the center where it will cut into the area that joins the shoulder and the shaft. Obliterate that area and the shoulder will fall off. 
The "milling" or "carving" bits of a dremel are also good at getting in and surgically taking out that kind of area in extractions like this.
What I usually do in situations like this is try to do most of the work with drill and dremel until I can tell I'm most of the way through, and set things up so I'm finishing by chiselling or wedging things apart manually with a hand tool. That way you give the part you're trying to save minimal and hopefully zero exposure to your cutting tools.
