Fluid trainer works with quick release rear axles, I have a standard nut fixie
Assuming you have the Cycle Ops Fluid 2 trainer, which is what I have, you can use it with nutted wheels.  The skewer that comes with it is just for bikes with fancy quick release skewers that don't fit into the trainer or if you have nice skewers that you don't want to scratch up.
For nutted wheels, the retaining mechanism should fit fine over top of the nuts.
Update
Thanks for the picture.
That isn't the Cycle Ops Fluid Trainer, so it looks like my previous info isn't really applicable.  Some trainers aren't designed to work with nutted wheels.  Looks like it's not only a problem with the way it mounts, but that your hub is too narrow.  The mechanism looks like it's screwed all the way in, but it's still too loose.  How much axle/bolt do you have sticking out from the nut? It looks like the part that's supposed go around the nut/skewer isn't very deep, and is probably meeting up with the end of the axle.  You could probably cut it shorter to alleviate this problem, but that's probably not an optimal solution.
There's a couple different adapters that I was able to find for using nutted axles in trainers linked below. It's difficult to tell if either one would work with your specific trainer.
Forza Bolt Axle Adapter
TACX axle nuts
Even if you do manage to get it mounted properly, it might not work that well. I mounted my fixie on my trainer once, and there wasn't enough resistance. Fluid trainers generally don't have adjustable resistance (maybe yours does) and required that you change gears to adjust the resistance. I found that the gear on my fixie (46x15, which is higher than most) wasn't hard enough to give me a decent workout on my trainer.