I used to do boxing earlier in college, and used to hit the gym religiously till last year, when my elbow fractured and got implants to hold it together. After that I wasn't able to do much activity, even if I wanted, apart from running. Even pushups carry a risk of screwing up the implants in my elbow.
What kind of physical activity do you think can be done in this type of scenario?
Maybe swimming? Possibly tennis or racquetball if you only use your other arm to hold the racquet. I'm not sure how easy it is to screw it up again, but if you can still handle running it seems like those two might be okay.
Pilates could be very useful. Most of it is very low impact strength training - instead of holding up your body weight, you're straining against springs, for example. There are a lot of exercises that don't focus on the arms because Pilates focuses on very discrete muscle groups for each exercise.
I wouldn't start with group classes, though: talk with your physical therapist (are you doing PT to help the recovery?) about appropriate exercises, and maybe sign up for a few introductory private pilates classes. Be sure to inform the instructor about the limitations you have.
It sounds like I have the same injury as you do. Five years ago I broke the olecranon in my left elbow and had to get it all screwed back together. To echo other suggestions, I would say swimming is definitely the way to go if you want to improve upper body fitness. Everything else I've found to be too high-impact and causes discomfort. Also keep up the running.
Some kind of casual low-impact sport (or at least low-bad-arm-impact), like pick-up soccer or disc golf. That way you'd still be getting the movement and coordination practice that you don't necessarily get from running. Also bicycling might give you the same benefits from running but with more new things to take in.
What kind of physical activity do you think can be done in this type of scenario?